Naturally light invisible axion and local Z13⊗Z3 symmetries

Abstract

We show that by imposing local Z 13 Z 3 symmetries in an SU(2) U(1) electroweak model we can implement an invisible axion in such a way that ͑i͒ the Peccei-Quinn symmetry is an automatic symmetry of the classical Lagrangian, and ͑ii͒ the axion is protected from semiclassical gravitational effects. In order to be able to implement such a large discrete symmetry, and at the same time allow a general mixing in each charge sector, we introduce right-handed neutrinos and enlarge the scalar sector of the model. The domain wall problem is briefly considered. DOI: 10.1103/PhysRevD.69.015007 PACS number͑s͒: 14.80.Mz, 11.30.Er, 12.60.Fr It has been known for a long time that the complex nature of the QCD vacuum solves the old U(1) A problem ͓1,2͔ but also implies the existence of the FF term in the classical Lagrangian ͓3͔. This CP-nonconserving angle ͑which has also an electroweak contribution͒ should be smaller than 10 Ϫ9 in order to be consistent with the measurement of the electric dipole moment of the neutron ͓4͔. Why is CP only weakly violated in QCD? This is the strong CP problem and an elegant way to solve it is by introducing a chiral symmetry U(1) PQ ͓5͔. This Peccei-Quinn ͑PQ͒ symmetry implies also the existence of a pseudo Goldstone boson-the axion ͓6͔. The invisible axion is almost a singlet ͑here denoted by ) under the SU(2) U(1) gauge symmetry ͓7͔. In addition, since this particle can occur in the early Universe in the form of a Bose condensate which never comes into thermal equilibrium, it can be a significant component of the cold dark matter if its mass is of the order of 10 Ϫ5 eV ͓8,9͔, which makes it important in its own right. However, such a small mass for the axion can be spoiled by semiclassical gravity effects since it could induce renormalizable or nonrenormalizable effective interactions which break explicitly any global symmetry, and in particular the PQ symmetry. The most dangerous of these effective operators are of the form D , where D is the mass dimension of the effective operator. In this case the axion could gain a mass which is greater than the mass coming from instanton effects and also the angle is not less than 10 Ϫ9 in a natural way. This can be avoided if the dimension of the effective operators is D տ12 ͓10,11͔. Hence, unless D is high enough, invisible axion models do not solve the strong CP problem in a natural way. Thus the invisible ͑very light͒ axion must be protected against gravity effects.Some years ago Krauss and Wilczek ͓12͔ pointed out that local symmetries can masquerade as discrete global symmetries to an observer equipped with only low-energy probes. We can see this as follows. Consider a U(1) gauge theory with two complex scalar fields and carrying chargeNe and e, respectively. It is possible, for instance, that undergoes condensation at some very high energy v, but does not condensate and produces quanta of relatively small mass. Before the condensation the theory is invariant under the transformation Ј(x)ϭexp͓iNe⌳(x)͔(x), Ј(x) ϭexp͓ie⌳(x)͔(x) and A Ј (x)ϭA (x)...

SU(5)⊗Z13grand unification model

“…[12], in such a way that the partner of the standard model fields in SU 5 multiplets are new heavy fields. This model allows a stable proton, unification of the three coupling constants, a natural PecceiQuinn symmetry of the classical Lagrangian, and the axion being protected against semiclassical gravity effects.…”

“…Finally, we add a 10 (D 10 ) and a 15 (T 15 ) which contains, respectively, the singlet h and the triplet T of Ref. [12]. T 15 gives Majorana masses to the active neutrinos.…”

“…[12] the representation content of the standard model was augmented by adding scalar fields and three right-handed neutrinos, in such a way that a discrete Z 13 Z 3 symmetry was implemented in the model there. Explicitly, the particle content of the model is the following: …”

“…[11] that in the context of the multi-Higgs extension of the standard model with an invisible axion proposed in Ref. [12] we have: (i) the unification of the three gauge coupling constants near the Peccei-Quinn scale; (ii) the model predicts the correct value of the weak mixing angle at the Z-peak; (iii) the axion and the nucleon are stabilized by the cyclic Z 13 Z 3 discrete symmetries; finally, (iv) although neutrinos got an arbitrary Dirac mass, through the effective…”

Neutrino mass through concomitant breakdown of the U(1) chiral and scale symmetries

“…One way out of this is to postulate gauge discrete Z N symmetries [8] giving some protection for the global factor U(1) P Q against gravity [9,10].…”