ar X iv : h ep - p h / 92 04 21 8 v 1 1 3 A pr 1 99 2 UTHEP - 233 April , 1992 On the Renormalization of the S Parameter

We calculate the S parameter of the standard model at one loop of fermions, using three different regularizations (dimensional, Pauli-Villars and lattice) and find an extra contribution to the S parameter besides the standard one for each case. This shows that the extra contribution recently reported for the lattice regularization is not necessarily tied to the non-decoupling effect of fermion doublers. We argue that the extra contribution should be subtracted in the renormalizable perturbative expansion. ⋄ email: [email protected] 1 A successful construction of chiral gauge theories on the lattice requires a dynamical decoupling of fermion doublers in the continuum limit. It now seems that in the WilsonYukawa formulation fermion doublers are indeed decoupled in the region of strong WilsonYukawa coupling . However, in a recent paper by Dugan and Randall, it is found that the S parameter calculated in the Wilson-Yukawa formulation has an extra contribution added to the standard continuum value. It is claimed that this extra contribution arises from fermion doublers, so that a decoupling of doublers does not hold in the Wilson-Yukawa formulation. In this paper, however, we point out that an extra contribution also arises in continuum regularization schemes. Therefore, an extra contribution is not necessarily tied to non-decoupling effects of fermion doublers. The S parameter is defined by S = −16π ∂Π3Y (p ) ∂p2 |p2=0 (1) where Π3Y (p ) is the part of the (T 3 L current) (Y current) correlation function proportional to δμν . For a quick review, see ref. 3. We consider a one-loop contribution of one fermion doublet to this S parameter in the standard model. At the fermion 1-loop level, for simplicity, we take the doublet Higgs field φ as φ(x) = g(x) ( 0 v ) where v is a vacuum expectation value of φ and gg = 1 . The Lagrangian for one fermion doublet in the standard model is given by LF = L0 + Lreg (2) where L0 = ψ(x) 4