Multiparticle processes and tamed ultraviolet divergences

A new approach to computing the amplitudes of multiparticle processes in renormalizable quantum field theories is presented. Its major feature is a separation of the renormalization from the computation. Within the suggested approach, new computational rules are formulated. According to the new rules, the amplitudes under computation are expressed as a sum of effective Feynman amplitudes of which the vertices are the complete amplitudes of the processes involving not more than four particles, and the lines are the complete two-point functions. The new rules include prescriptions for computing the combinatorial factors by each amplitude. It is demonstrated that, due to these prescriptions, the combinatorial factors by the amplitudes that are divergent in the ultraviolet in four space-time dimensions vanish. Because of this, the computations within the new approach do not involve the ultraviolet renormalization.

Figure 1

a: The tree corresponding to the double partition $t=\{\{2,2,1\},\{1\}\}$; dark edges are related to the expansion of the logarithm, and the light ones are related to the expansion of the exponential of Eq. (18). b: A complete tree extending the tree on the left.

Figure 2

Graphs and threes of the third order; numbers on the graph panes are $|Aut(g)|$ of Eq. (41) and on the tree panes are the $Z(t)$ factors of Eq. (40).