It is remarked that left-right symmetry of the starting gauge interactions is a "natural" symmetry if it is broken in no way except possibly by mass terms in the Lagrangian. The implications of this result for the unification of coupling constants and for parity nonconservation at low and high energies are stressed.In a recent note1 it has been pointed out that a left -right discrete symmetry2 in the starting gauge interactions provides a desirable basis f o r introducing CP violation into such a scheme, in that such a gauge symmetry, subjected to spontaneous symmetry breaking, automatically satisfies the relation q+-=qo0 and links the suppression of CP violation to the known suppression of (V + A ) weak interactions. The purpose of this note is to point out that such a discrete symmetry h a s a second desirable feature in that it can indeed be preserved a s a "natural" symmetry, provided we ensure that the symmetry in question is broken, if a t all, only in a "soft" manner, i.e., via m a s s t e r m s of the Higgs s c a l a r s .We point out that such a soft symmetry breaking is, on the one hand, sufficient to guarantee the desired m a s s splittings between the left-and the right-handed gauge bosons (with rn,, >> m,,), s o a s to account for parity nonconservation and dominant (V -A ) character of the known weak interactions. On the other hand, it does not lead to infinite c o rrections in higher o r d e r s to the difference between the left-and the right-handed gauge coupling constants gL and g,, s o that (g, -gR) i s calculable.This in turn h a s important implications: (1) A unifying gauge model of the type S U (~) , X S U (~) , based3 on 2n four-component fermions (which may comprise gauge groups of Ref. 2 a s subgroups), subjected to a discrete left -right symmetry a s above, can be described by a single coupling cons t a~t ,~ a feature which i s desirable from the point of view of unification of all forces and all matter.(2) Parity nonconservation (like CP nonconservation) can be interpreted a s a low-energy phenomenon to disappear a t high energies, a s conjectured in Refs. 1 and 2. We also briefly discuss the implications of such a model f o r neutral-current phenomenology. To make our discussions specific, let u s assume, f o r illustration only, that the gauge group i s of the f o r m where SU(2), generates (V -A ) known weak interactions, SU(2), generates parallel (V + A ) interactions in the manner suggested in Ref. 2, while G denotes any gauge group commuting with SU(2), xSU(2), and generating gauge interactions with L-R symmetry. [~x a m p l e s~ of G a r e SU(~)'$; o r SU(~)?:; XU(~),+,.] We will assume that the bare coupling constants gp' and g$' for the gauge groups SU(2), and SU(2), a r e equal; i n this c a s e the gauge interactions generated by the entire group 9, a r e L -R symmetric. We then show that the renormalized coupling constants g, and gR a r e equal up to finite radiative corrections a s long a s the L a R symmetry is not broken by the Higgs potential in a hard manner. [~t will ...
