Distributional limits of renormalized Feynman integrals with zero-mass denominators (Q1224288)

Within the Bogolyubov--Parasyuk--Hepp--Zimmermann framework of renormalized perturbation theory, the connected Green functions of quantum fields are expressed as sums of contributions from Feynman diagrams, each of which corresponds to an integral \[ T_\varepsilon(p)=\int d^{4m}k R_\varepsilon(p_1,\dots,p_n; \, k_1,\dots,k_m). \] The authors show that if the integral \[ T_\varepsilon(\varphi)=\int d^{4m}k d^{4n}p\varphi(n)R_\varepsilon(p;k) \] converges absolutely for every \(\varepsilon >0\) and \(\varphi\) in the Schwartz space of test functions \(\mathcal S(\mathbb R^{4n})\), then \(T_\varepsilon\) approaches, when \(\varepsilon\) tends to zero, a Lorentz invariant limit as a tempered distribution. This theorem generalizes previous results of Hepp and Zimmermann to cases where some or all of the mass parameters vanish.