Universal Seesaw from Left-Right and Peccei-Quinn Symmetry Breaking

To generate the lepton and quark masses in the left-right symmetric models, we can consider a universal seesaw scenario by integrating out heavy fermion singlets which have the Yukawa couplings with the fermion and Higgs doublets. The universal seesaw scenario can also accommodate the leptogenesis with Majorana or Dirac neutrinos. We show the fermion singlets can obtain their heavy masses from certain global symmetry breaking, which is driven by one complex scalar singlet or two. The global symmetry can be identified to the Peccei-Quinn symmetry since it is mediated to the standard model quarks at tree and/or loop level.

Introduction: In the SU (3) c × SU (2) L × U (1) Y standard model (SM), the charged fermion singlets have Yukawa interactions with the fermion and Higgs doublets so that the quarks and the charged leptons can obtain their Dirac masses after the Higgs doublet develops its vacuum expectation value (VEV). However, we can not get the neutrino masses in this way because the righthanded neutrinos are absent in the SM. To naturally generate the neutrino masses, which are far lower than the charged fermion masses, we can consider the seesaw [1] extension of the SM by introducing the right-handed neutrino singlets with heavy Majorana masses [1] and/or the left-handed Higgs triplet(s) with small VEV(s) [2].
The SM and its seesaw extension can be embedded in the SU (3) c ×SU (2) L ×SU (2) R ×U (1) B−L left-right symmetric models [3], where the left(right)-handed fermions are placed in SU (2) L [SU (2) R ] doublets. For the breakdown of SU (2) R ×U (1) B−L to U (1) Y , the simplest choice is to introduce a right-handed Higgs doublet and its lefthanded partner. The fermion doublets associated with the additional fermion singlets can have the Yukawa couplings with the Higgs doublets [4][5][6][7]. Subsequently we can integrate out the fermion singlets to generate the masses of the quarks, the charged leptons, and the neutral neutrinos. In this scenario, the seesaw is a universal origin of the fermion masses. The universal seesaw can also accommodate the leptogenesis [8] mechanism with Majorana [8] or Dirac [9] neutrinos to explain the matterantimatter asymmetry in the universe. The key point of the universal seesaw is the existence of the heavy fermion singlets, including the color singlets for generating the lepton masses and the color triplets for generating the quark masses. The heavy masses of the fermion singlets can be simply input by hand as they are allowed by the gauge symmetry. A more interesting possibility is that the fermion singlets obtain their masses through certain spontaneous symmetry breaking. For example, the original work on the universal seesaw introduced a new SU (3) gauge symmetry [4].
In this paper we shall consider a spontaneous Peccei-Quinn (PQ) symmetry [10] breaking to generate the heavy masses of the fermion singlets for the universal seesaw. Specifically we shall impose a global symmetry under which the left-and right-handed fermion singlets carry equal but opposite charges. Accordingly, the left-and right-handed fermion and/or Higgs doublets also carry equal but opposite charges through their Yukawa couplings with the fermion singlets. In this context, the left-right symmetry should be the chargeconjugation. The spontaneous symmetry breaking of the global symmetry is driven by one complex scalar or two. The fermion singlets can obtain the heavy masses through their Yukawa couplings with the complex scalar singlet(s). The Nambu-Goldstone boson (NGB), associated with the global symmetry breaking, can become a pseudo NGB (pNGB) as it picks up a tiny mass through the color anomaly [11]. Since the global symmetry is mediated to the SM quarks at tree and/or loop level, it can be identified with the PQ symmetry to solve the strong CP problem. Consequently, the pNGB should be an invisible [12,13] axion [10,14].
We emphasize that with the Yukawa couplings (6), the strong CP phase will not vanish at tree level. The universal seesaw models can provide a solution to the strong CP problem without an axion [16] if the left-right symmetry is not the charge-conjugation but the parity [17]. However, the parity as the left-right symmetry is inconsistent with the global symmetry (4), which is essential for the mass generation of the fermion singlets.
Universal seesaw : By integrating out the charged fermion singlets, D L,R , U L,R , and E L,R , we can have the following dimension-5 operators, After the left-right symmetry breaking, the right-handed charged fermions can obtain their Yukawa couplings with the left-handed fermion and Higgs doublets. Such Yukawa couplings, identified to those in the SM, can generate the Dirac masses of the charged fermions.
In the neutrino sector, the neutral fermion singlets N L,R can form the Majorana or pseudo-Dirac fields, depending on the size of M D N and M M N . In the pseudo-Dirac case with M D N ≫ M M N , the induced dimension-5 operators are which can account for the light Dirac neutrinos. In the Clearly, the right-handed neutrinos can get a Majorana mass term from the operators involving the right-handed doublets. Their Yukawa couplings to the left-handed lepton and Higgs doublets can be induced by the operators involving the left-and right-handed doublets. This means we have realized the type-I seesaw [1]. Furthermore, the operators involving the left-handed doublets will give an additional contribution to the neutrino masses, playing a role of the type-II seesaw [2].

Peccei-Quinn symmetry:
We can classify our model into three cases by choosing the quantum numbers of the fermion and Higgs doublets, i.e.

The Higgs doublets are trivial under the global
symmetry. In this case, we have x 1 = x 2 as y = 0 but z = 0. Thus we only need one complex scalar singlet, i.e. ξ 1 = ξ 2 .
2. The fermion doublet are trivial under the global symmetry. In this case, we have x 1 = −x 2 as z = 0 but y = 0. There is only one complex scalar singlet, i.e. ξ 1 = ξ * 2 . 3. The fermion and Higgs doublets are not trivial under the global symmetry. In this case, the quantum numbers x 1 and x 2 keep independent as y = 0 and z = 0. This means the existence of two complex scalar singlets.
In the following we will clarify the global symmetry in any cases is identified with the PQ symmetry.
In the first case, the unique complex scalar singlet can be described by with a being the NGB. It is easy to derive the tree-level couplings of the NGB to the colored fermions, (12) where Q denotes D and U while q stands for the SM quarks. The first term of the above Lagrangian associated with the tree-level Q − q mixing will result in With the quark-gluon interactions, it will also induce at two-loop order, as in the Kim-Shifman-Vainshtein-Zakharov [12] (KSVZ) model. Clearly, the couplings of the NGB to the SM quarks are dominated by Because of the instanton interaction, the NGB can pick up a tiny mass [14,18], where N = 3 for three families of the SM quarks while Z ≃ m u /m d .
In the second case, we also have one complex scalar singlet, The couplings of the NGB to the colored fermions should be Similar to Eqs. (13) and (14), we compute the induced couplings to the SM quarks, where c q = 1 for u, c, t while c q = −1 for d, s, b. The mass of the NGB is given by In the third case, there are two complex scalar singlets, The couplings of the two NGBs to the colored fermions are given by Clearly, this case is identified to the first or second case for x 1 = ±x 2 . So, let's consider x 1 = ±x 2 . In this case we can neglect the contributions from the colored fermion singlets, The two NGBs thus get their tiny masses as below, In the above three cases, the NGBs from the global symmetry breaking all couple to the axial vector current of the SM quarks and hence obtain the instanton-induced masses. So, the global symmetry is the PQ symmetry as the pNGB acts as the axion. For convenience, we express the axion mass as where the axion decay constant f a can be derived from Eq. (16), (20) or (24). The PQ symmetry breaking scale should be high enough to fulfill the theoretical and experimental constraints [19]. For example, the PQ symmetry breaking may happen before inflation to avoid the cosmological domain wall problem. With an appropriate PQ symmetry breaking scale, the axion can act as the dark matter. Leptogenesis: After the left-right symmetry breaking, the leptogenesis can be applied to explain the matterantimatter asymmetry in the universe. If the neutral fermion singlets and hence the neutrinos form the pseudo-Dirac fields, the decays of the neutral fermion singlets can produce a lepton asymmetry stored in the left-handed lepton doublets and an equal but opposite lepton asymmetry stored in the right-handed neutrinos. Since the effective Yukawa interactions between the leftand right-handed neutrinos are extremely weak, they will go into equilibrium at a very low temperature where the sphaleron [20] action is not active. Therefore, the sphaleron process can partially transfer the left-handed lepton asymmetry to a baryon asymmetry. This leptogenesis scenario with Dirac neutrinos is titled as neutrinogenesis [9]. In the other case with the neutral fermion singlets being the Majorana fields, we can have the conventional leptogenesis with Majorana neutrinos.
Conclusion: In this paper we connected the universal seesaw scenario, where not only the neutral neutrinos but also the charged fermions obtain their masses through the seesaw mechanism, to the PQ symmetry for solving the strong CP problem. In our model, the fermion singlets, including the color triplets for generating the quark masses and the color singlets for generating the lepton masses, have the Yukawa couplings to one complex scalar singlet or two. The scalar singlet acquires a large VEV to spontaneously break the global PQ symmetry. So, the fermion singlets can obtain their heavy masses for the realization of the universal seesaw. The colored fermion singlets mediate the PQ symmetry to the SM quarks at tree and/or loop level so that the axion can pick up a tiny mass through the color anomaly. We thus naturally related the PQ symmetry to the neutrino mass-generation [21]. Our model also accommodates the leptogenesis with Majorana or Dirac neutrinos.
ML is supported by the Sonderforschungsbereich TR 27 of the Deutsche Forschungsgemeinschaft. PHG is supported by the Alexander von Humboldt Foundation.