Infinite renormalization of θ-term and Jarlskog invariant for CP -violation


The logarithmic ultraviolet divergence in the θ-term induced by electroweak interactions in the Kobayashi-Maskawa model of CP -violation, found by Ellis and Gaillard, is discussed. We relate it to the Jarlskog CP -odd invariant of quark mass matrix. The divergence arises in 14th order in Higgs coupling as well as in 12th order plus 2nd order in U(1) gauge boson coupling. e-mail address: KHRIPLOVICH@INP.NSK.SU e-mail address: VAINSHTEIN@MNHEP.HEP.UMN.EDU The purpose of this note is to comment on the connection between the infinite renormalization of the θ-term found by Ellis and Gaillard[1] in the Kobayashi-Maskawa (KM) model of CP -violation and the Jarlskog invariant[2] of quark mass matrix for CP -violating effects in this model. It was first noted by Wilczek[3] that the parameter θ in the structure θ g S 32π GaμνG̃ a μν (1) can get infinitely renormalized due to CP -violating electroweak interactions. Then it was shown in ref.[1] that such infinite renormalization can occur only starting from 14th order in the weak coupling gW . It looks only natural that operator (1) of dimension 4 gets infinitely (logarithmically) renormalized. Amazing is such a high order of perturbation theory where the infinite renormalization shows up. On the other hand due to the GIM mechanism the very existence of this ultraviolet logarithmic divergence may look strange. Indeed in a renormalizable gauge the ultraviolet divergence of any loop contributing to the operator (1) cannot be stronger than logarithmic one. One may expect that the GIM mechanism leads to cancellation of logarithmic divergencies in the θ-term. Such an improvement of the ultraviolet behaviour takes place indeed. In particular the order g W g 2 S correction to the θ-term turns out to be finite even in the local limit of the four-fermion interaction[4]. The crucial observation of ref.[1] is that the θ-term divergence arises in a renormalizable gauge not due to the W-exchange, but due to the Higgs one. The Higgs coupling with fermions has the form φ (D̄LHD DR + ŪR H † U UL) + φ + (ŪL HD DR − ŪR H † U DL) + h.c. (2) Here φ, φ are the Higgs field components,

Cite this paper

@inproceedings{Khriplovich1993InfiniteRO, title={Infinite renormalization of θ-term and Jarlskog invariant for CP -violation}, author={I . B . Khriplovich and A . I . Vainshtein}, year={1993} }