Loop corrections to the four-fermion interaction

Abstract We consider the $$f f \rightarrow f f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mi>f</mml:mi> <mml:mo>→</mml:mo> <mml:mi>f</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> scattering amplitudes for the a massless four-fermion interaction model in four dimensions. At first we take the simplest version with the scalar current–current interaction. The loop corrections up to the three-loop level are calculated within the spinor-helicity formalism using the Weyl spinors. We find out that there are two independent spinor structures that appear in all orders of perturbation theory that can be separated when calculating the diagrams. Our aim is to calculate the leading divergences within the dimensional regularization. To check the validity of our calculations, we use the recurrence relations that connect the leading divergences in the subsequent orders of perturbation theory. We left the derivation of these relations and further analysis of the consequences for another publication for the sake of clarity of the current presentation. At the end we briefly consider the physically more interesting case of V–A interaction.