Infrared and ultraviolet divergences of the coefficient functions of Feynman diagrams as tempered distributions. I

Conditions of the presence of infrared and ultraviolet divergences of coefficient functions corresponding to arbitrary scalar Feynman diagrams and considered as tempered distributions are found. Analytical regularisation is used to analyse both types of divergences. It is shown that for any graph $\Gamma$ there is a domain of regularising complex parameters $\lambda_l$ in which the corresponding coefficient function is an analytical function of these parameters (in the distribution theory sense) possessing analytical continuation into all of $C^{\mathscr L}$ as a meromorphic function with two series of poles (“ultraviolet” and “infrared” ones). Infrared poles are located on hyperplanes defined by relationships: $\sum_{l\in\gamma}\lambda_l=-\Omega^\Gamma(\gamma)+n$, $n=0,1,\dots$ and $\Omega^\Gamma(\gamma)$ being the index of infrared divergency of a subgraph $\gamma$ of the graph $\Gamma$. These relationships are to be written for the graphs including massless particles only.