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Rescaling mechanism and effective symmetry from the ideal cancellation of the $S$ parameter in custodial models

We present a model independent analysis of custodial corrections to the $S$ parameter. The negative contributions coming from direct corrections, i.e. the corrections associated to effects of new physics in the fermionic sector, can be used to eliminate unwanted positive oblique contributions to $S$. By means of such an ideal cancellation among oblique and direct corrections the electroweak physics can be made insensitive to custodial new physics. The Lagrangian analysis of the ideal cancellation reveals a possible sizable redefinition of the effective electroweak energy scale with respect to models with only oblique corrections. We then infer from general principles the full expression for the effective custodial operator responsible in the gauge sector for the contribution to $S$. We show that the ideal cancellation exactly eliminates all the custodial corrections, including those in the non-abelian and longitudinal terms. Indeed, provided redefinitions of the physical parameters, the standard model Lagrangian can be equivalently defined modulo custodial corrections in the gauge and fermionic sectors. Thus the ideal cancellation can be regarded as an effective symmetry. Finally we investigate the theoretical origin of this effective symmetry in terms of the effective gauge structure induced by custodial new physics.