# Hadamard expansion of interacting Klein Gordon 2-point function

Context: There is an algorithm due to Hadamard, I believe, for constructing local bi-distributional solutions to elliptic and hyperbolic equations for the purpose of proving existence and uniqueness results for such equations (see: Garabedian, P.R., Partial differential equations, New York-London-Sydney: John Wiley and Sons, Inc. XII, 672 p. (1964). ZBL0124.30501.).

This algorithm allows one, e.g., to construct a principal (symmetric) solution, $H(x,x')$, to the Klein-Gordon wave operator $\square_x\equiv g_{ab}\nabla^b_x\nabla^a_x-m^2$ in both variables, where $g_{ab}$ is a Lorentzian-signature metric and $\nabla_a$ is the covariant derivative. For even dimensions $D\ne2$, the algorithm proceeds by taking the principal solution to be of "Hadamard form",

$$H(x,x')\equiv \frac{U(x,x')}{\sigma^{D/2-1}(x,x')}+V(x,x')\ln\sigma(x,x')+W(x,x'), \label{H} \tag{*}$$ where $U(x,x')\equiv\sum^{D/2-2}_{i=0}U_i(x,x')\sigma^i$, $V(x,x')\equiv \sum^\infty_{i=0}V_i(x,x')\sigma^i$, $W\equiv \sum^\infty_{i=0} W_i(x,x')\sigma^i$, and $\sigma(x,x')$ is the squared geodesic distance between points $x$ and $x'$ which are assumed to lie in a convex normal neighborhood with a unique geodesic connecting them. Remark: If one makes an "infinitesimal shift" $\sigma\to \sigma+2i\epsilon(t-t')+\epsilon^2$, then $H(x,x')$ is said to have "singularity structure consistent with the definition of the Feynman propagator as a time-ordered product".

One then substitutes \eqref{H} into the K-G equation in $x$ (at fixed $x'$) and equates coefficients of the explicitly appearing powers of $\sigma$ and $\ln \sigma$ to zero. In this way, one obtains a sequence of ordinary differential equations for $U_i$, $V_i$, and $W_i$ along the geodesic connecting $x$ and $x'$. The equations for $U_i$ and $V_i$ may be solved recursively and have a unique regular solution expressed in terms of integrals along the geodesic connecting $x$ and $x'$. [See this paper for the explicit recursion relations for the K-G theory]

Question: I have seen reference to algorithms for computing power series solutions for nonlinear differential equations--e.g., the Parker-Sochacki method. I am asking is the analogue of $H(x,x')$ known, e.g., for $\phi^4$-theory? i.e., what is the form of the principal solution for the nonlinear operator of the form $\square_x-\frac{1}{4!}\phi^3(x)$ expressed as an asymptotic short-distance expansion? Or, more generally, what methods would be applicable to theories of the form $\square_x - \alpha_i\phi^i(x)$ for $i\in\mathbb{Z}$?