High-energy behaviour of Fermi theory

Abstract We consider the 4-fermion scattering amplitude in massless Fermi theory. Based on the Bogolyubov–Parasyuk theorem, which guarantees locality of the counter terms, we derive the recurrence relations for ultraviolet divergences of diagrams that establish a connection between successive orders of perturbation theory. We check their validity up to three loops comparing them with explicit calculation made earlier. Then we construct the corresponding RG equation that sums up the leading logarithmic contributions in all orders of perturbation theory. Numerical analysis of these equations in the asymptotic regime $$s\sim t\sim u \sim E^2 \rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>∼</mml:mo> <mml:mi>t</mml:mi> <mml:mo>∼</mml:mo> <mml:mi>u</mml:mi> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> is performed for two cases: the unit and the V – A operator in the fermion current. We found out that for the unit operator the high energy behaviour of the theory in the leading order is characterized by the presence of the Landau pole, while for the V – A operator the theory is asymptotically free. Therefore, in the latter case, radiative corrections restores unitarity, which is violated at the tree level. We compare the obtained behaviour of the amplitude with one in the theory with the intermediate gauge bosons and found an overlap between them.