Natural Inflation and Flavor Mixing from Peccei-Quinn Symmetry Breaking

We propose a left-right symmetric model to simultaneously give natural inflation and flavor mixing from a Peccei-Quinn symmetry breaking at the Planck scale. Our model can be embedded in SO(10) grand unification theories.

In this paper we propose a novel left-right symmetric scenario where the PQ symmetry is realized such that it naturally leads simultaneously to inflation [14] and flavor mixing. The specific model which we will discuss can be embedded in SO(10) GUTs, where the particle content emerges from a bigger picture. However, we will not discuss this embeddings further and we will focus on the main mechanism within the left-right framework. At the left-right level our model contains one Higgs bi-doublet for each family, two Higgs doublets, two leptoquark doublets and six complex singlets in the scalar sector while three neutral singlets and three generations of lepton and quark doublets in the fermion sector. The six scalar singlets are responsible for a U (1) 6 global symmetry breaking at the Planck scale. Because of the Yukawa interactions between the scalar and fermion singlets, the U (1) 6 symmetry is explicitly broken down to a U (1) 3 symmetry [15]. Three Nambu-Goldstone bosons (NGBs) will obtain heavy masses through the Coleman-Weinberg potential [5] while the other three will pick up tiny masses through the color anomaly [16]. The heavy and light pNGBs can act as the inflaton and the axion, respectively. This inflationary scenario can also avoid the cosmological domain wall problem [17]. In the absence of any off-diagonal Yukawa couplings involving the lepton and quark doublets, we can make use of the mixed fermion singlets to induce the lepton mixing by tree-level seesaw [18] and the quark mixing by one-loop diagrams.
The Model : The scalar sector includes one Higgs bidoublet for each family, two Higgs doublets, two leptoquark doublets, and six complex singlets, In the fermion sector, there are three neutral fermion singlets, and three generations of quark and lepton doublets, We assume a discrete left-right symmetry which is connected to charge-conjugation and the fields thus will transform as In the presence of the above left-right symmetry, we can impose a family symmetry U (1) F = U (1) 3 , under which the left-and right-handed fermion doublets carry an equal but opposite charge for each family, i.e.
We also assign the U (1) F charges for other fields, We further introduce a global symmetry U (1) G , under which the fields carry the following quantum numbers, Next we specify the allowed Yukawa interactions, For simplicity, we do not write down the full scalar poten- should be absent due to the global symmetry U (1) G . Instead, we only give the part relevant for generating the mixing between the left-and right-handed leptoquarks, Clearly, the lepton and quark doublets, the Higgs and leptoquark doublets, the Higgs bi-doublets, the fermion singlets and the scalar singlets can, respectively, belong to the 16 F i , 16 H , 10 H i , 1 F i and 1 H ij representation in SO(10) GUTs.
Pseudo Nambu-Goldstone bosons: Each scalar singlet σ ij has an independent phase transformation to perform a U (1) 6 symmetry. However, the six scalar singlets have the Yukawa interactions with the three fermion singlets [the g-terms in Eq. (11)] so that the U (1) 6 symmetry should be explicitly broken down to a U (1) 3 symmetry. After the six scalar singlets develop their vacuum expectation values (VEVs), there will be six NGBs, i.e.
The fermion singlets then obtain their Majorana masses which will result in a Coleman-Weinberg potential [5,15], with Λ being the ultraviolet cutoff.
Only three NGBs can exist in the Coleman-Weinberg potential while the other three can be absorbed by the three fermion singlets. For example, we can take Clearly, we have ϕ ′ ij = 0 for i = j. By taking a reasonable simplification on the logarithm A combination of ϕ ′ 12 , ϕ ′ 13 and ϕ ′ 23 can have a potential of the form as below, The pNGB ϕ can realize the natural inflation for µ = O(10 15 GeV) and f = O(M Pl ) [14]. The Majorana masses M ij should be determined by the Yukawa couplings g ij = O(10 −4 ) for the given symmetry breaking On the other hand, the Yukawa interactions (11) mean that the three NGBs absorbed in the three fermion singlets can have derivative couplings with the quarks, Therefore, the NGBs ϕ ii can obtain their tiny masses through the color anomaly. Clearly, the pNGBs ϕ ii play the role of the invisible axion [19,20] while the family symmetry U (1) F is identified with the PQ symmetry. Benefited from the inflation, our model can escape from the cosmological domain wall problem [17]. Furthermore, for the PQ symmetry breaking at the Planck scale, we can choose the initial value of the misalignment angle by the anthropic argument [21,22] to give a desired dark matter relic density [3,4]. The mass terms involving neutral leptons are given by For f χ 0 R an/orM much bigger thanm ν and f χ 0 L , we can make use of the seesaw formula [11] to derive the neutrino masses, i.e.
Here the first term is the double seesaw [12] forM ≫ f χ 0 R or the inverse seesaw [23] forM ≪ f χ 0 R while the second term is the linear seesaw [13], as shown in Fig. 1. With a left-right symmetry breaking scale χ 0 R = O(10 13 GeV), the double/inverse seesaw term should be the double seesaw forM = O(10 15 GeV). In this scenario, the inflaton should decay into the righthanded neutrinos through the off-shell fermion singlets. Subsequently, the decays of the right-handed neutrinos can realize the non-thermal or thermal leptogenesis [24].

FIG. 2: One-loop diagrams for quark masses and mixing.
Since the charged lepton mass matrix in Eq. (21) has a diagonal structure, the neutrino mass matrix (23) should account for the lepton mixing described by the PMNS matrix. With the six elements in the Majorana mass matrixM , we have enough flexibility to fit the known masses and mixing [18].
Quark masses and mixing: At tree level the mass matrices of the down-and up-type quarks are both diagonal, and (m 0 At one-loop order the leptoquarks can mediate the mixing of the fermion singlets to the quark sector, as shown in Fig. 2.
To calculate the loop-induced quark masses, we define the mass eigenstates of the leptoquarks, with the rotation angles, θ 2/3 ∈ [0, π] for δ 2 2/3 = 0 or θ 2/3 = π 4 for δ 2 2/3 = 0 ,(26a) θ 1/3 ∈ [0, π] for δ 2 1/3 = 0 or θ 1/3 = π 4 for δ 2 1/3 = 0 ,(26b) and the masses, Here we have defined We further rotate the fermion singlets to diagonalize their Majorana mass matrix, We then give the loop-induced quark masses as below, It is easy to check that the tree-level and loop-order quark mass matrices can induce the desired quark masses and mixing for a proper parameter choice. Summary: We discussed a left-right symmetric model with PQ symmetry with the aim to combine in a natural way inflation and flavor mixing. A key ingredient is the Yukawa interactions between the six scalar singlets and the three fermion singlets which explicitly break a U (1) 6 symmetry to a U (1) 3 symmetry. Among the six NGBs, three can obtain heavy masses through the Coleman-Weinberg potential to drive the natural inflation while the other three can pick up tiny masses through the color anomaly to solve the strong CP problem and to explain dark matter. Although the PQ symmetry forbids any offdiagonal Yukawa couplings involving lepton and quark doublets, we can mediate the mixing of the fermion singlets to the neutrino sector by tree-level seesaw and to the quark sector by one-loop diagrams. Our model can be embedded in SO(10) GUTs, but we did not discuss the details which do not change the proposed mechanism.