Binary icosahedral flavor symmetry for four generations of quarks and leptons Open Access

Abstract

1. Introduction

The current version of the standard model (SM) consists of three generations of quarks and leptons. Recently we proposed [1] a four generation lepton model based on the non-Abelian discrete symmetry |$A_{5}\equiv I$| (icosahedron), in which the best features of the three family |$A_{4}\equiv T$| (tetrahedral) model survive. Besides the new heavy degrees of freedom in the A₅ model, which satisfy the experimental constraints, we retain tribimaximal neutrino mixings, three light neutrino masses, and three SM charged lepton masses in the three light generation sector.

In this paper, we will explore a generalization of our |$A_5$| model to include four generations of both quarks and leptons. But before launching into that discussion we must first discuss the viability of models with four generations, given recent experimental developments. A fourth generation is now being constrained [2–5] by precision electroweak data [6], by flavor symmetries [7], and by the Higgs-like particle at 125 GeV recently reported at the LHC [8–10]. The new data provide an important step forward in distinguishing various four generation models, and in particular eliminating some from consideration. In particular, four sequential generation models are now highly disfavored [2–5]; however, it would be premature to dismiss all four generation models. While tension between four generation models and data has developed, a fourth generation is not excluded by the electroweak precision data [11–15], so the existence of a fourth generation is still a viable phenomenological possibility that can provide an explanation of the observed anomaly of CP asymmetries in the B meson system [16–18], and the baryon asymmetry of the universe [19,20], with additional mixings and CP phases. Also, there are a number of way to relieve this tension. For example, two Higgs doublet models (see, e.g., Ref. [21] and references therein) can accommodate a fourth generation of fermions and current data. For a comprehensive review see Ref. [22]. Typically these two Higgs doublet models are low energy effective field theories that require composite Higgses similar to top quark condensate models [23]. For some recent examples see Refs. [24–27]. Another possibility is to add electroweak doublets that are in color octets [28]. Further discussion can be found in Ref. [29]. While the model we will discuss has an extended Higgs structure, a full exploration of the possible composite nature of the scalar sector is beyond the scope of our present study.

To generalize our |$A_5$| model to include four generations of quarks and leptons, we first recall the three family scenarios in which the binary tetrahedral group |$T'\equiv SL_{2}(3)$| is capable of providing a model of both the quarks and leptons with tribimaximal mixings and a calculable Cabibbo angle [30]. The T′ group is the double covering group of |$A_{4}$|⁠. It has four irreducible representations (irreps) with identical multiplication rules to those of |$A_{4}$|⁠, one triplet 3 and three singlets |$\textbf {1}, \textbf {1}'$|⁠, and |$\textbf {1}^{\prime \prime }$|⁠, plus three additional doublet irreps |$\textbf {2}, \textbf {2}'$|⁠, and |$\textbf {2}^{\prime \prime }$|⁠. The additional doublets allow the implementation of the |$\textbf {2}\oplus \textbf {1}$| structure to the quark sector [31–36]; thus, the third family of quarks are treated differently and are assigned to a singlet. Hence they can acquire heavy masses [37–39]. One should note that |$A_{4}$| is not a subgroup of T′; therefore, the inclusion of quarks into the model is not strictly an extension of |$A_{4}$|⁠, but instead replaces it [40]. Based on the same philosophy, we study the model of four families of quarks and leptons by using the binary icosahedral group |$I'\equiv SL_{2}(5)$|⁠. The relation between I′ and A₅ is similar to that for T′ and |$A_{4}$|⁠. The icosahedral group |$A_{5} \subset SO(3)$| has double-valued representations that are single-valued representations of the double icosahedral group |$I' \subset SU(2)$|⁠. Hence, besides the irreps of I′ that are coincident with those of A₅, there are four additional spinor-like irreps |$\textbf {2}_{s}, \textbf {2}'_{s}, \textbf {4}_{s}$|⁠, and |$\textbf {6}_{s}$| of I′. We shall be able to assign quarks to the spinor-like representations, but to discuss model building using I′, we must first review our lepton model based on A₅, which will remain essentially unchanged when generalized to I′. Some useful group theory details have been relegated to the Appendix.

2. The leptonic A₅ model

The irreps of A₅ are one singlet 1, two triplets 3 and |$\textbf {3}'$|⁠, one quartet 4, and one quintet 5. The model is required to be invariant under the flavor symmetry of |$A_{5}\times Z_{2}\times Z_{3}$| and the particle content is given by Table 1.

3. I′ symmetry and the quark sector

The irreps of I′ are one singlet 1, two triplets 3 and |$\textbf {3}'$|⁠, one quartet 4, and one quintet 5, which are also the irreps of A₅, plus I′ has four spinor-like irreps |$\textbf {2}_{s}, \textbf {2}'_{s}, \textbf {4}_{s}$|⁠, and |$\textbf {6}_{s}$|⁠. The characters and the multiplication rules of I′ symmetry can be found in Table 2 and Table 3 of the Appendix.

4. Masses and mixings

Recall that, in the lepton A₅ model, H₄ is responsible for the Dirac masses of neutrinos, and we require the condition |$\langle H_{1}\rangle \equiv V_{1} \gg \langle H_{3_{1,2,3}}\rangle \equiv V_{3_{1,2,3}}$| in order to decouple the fourth generation neutrino from the three light SM neutrinos. Therefore, from |$M_{t,t'}$|⁠, we obtain the masses of |$t$| and |$t'$| as |$m^2_{t,t'} \approx \left [V^2_{1} + (V^2_{3_1} + V^2_{3_2} + V^2_{3_3}) \mp V_{1} \sqrt {4V^2_{3_1} + 2(V_{3_2} + V_{3_3})^2}\right ]/2$|⁠. For |$M_{bb'}$|⁠, we also follow the lepton |$A_5$| model by taking⁴|$\langle H'_{4}\rangle = (V'_{1}, V',V',V')$|⁠, since this gives masses to charged leptons too. |$m^2_{b,b'}$| are then given by |$[V^{\prime 2}_{1} + 3V^{\prime 2} \mp 2\sqrt {3}V'_{1}V']/{2}$|⁠. In general, we have enough parameters to accommodate the heavy quark mass spectrum.

5. Conclusion

As mentioned in the introduction, the recent observations of a boson with a mass near 125 GeV [8,9] have placed severe constraints on the standard model augmented by a sequential fourth generation of fermions. The CMS experiment has now excluded such a fourth generation of fermions with masses of up to 600 GeV [10]. Note that our I′ model has three doublets and one singlet Higgs, all of which are in non-singlet irreps of I′. Hence it is not necessarily disfavored by the current experimental search. Indeed, such severe limits can be relaxed into the range of 400–600 GeV in a two Higgs doublets model with four generation of fermions, as discussed in Ref. [22]. It would be interesting to investigate whether our I′ model (or one of its extensions) can be recast into a form designed in Ref. [22].

In summary, we construct a model of four fermion generations based on the binary icosahedral symmetry group I′. Many properties of the SM with three families are accommodated, such as the mass spectrum, tribimaximal mixings in the neutrino sector, and an identity CKM matrix at leading order⁵ . In addition, quark and lepton relations are intimately connected as their masses are provided from the same set of scalars. We believe that this makes the model both interesting and challenging. For example, one has to strike a balance between the result of tribimaximal mixings in the neutrino sector and derivation of a realistic Cabibbo angle from perturbations in the quark sector. It is still not clear to us whether the higher order corrections will lead to a realistic Cabibbo angle or if we need extra degrees of freedom to realize it. We will leave this and other phenomenological aspects of this model to future work.

Acknowledgements

This work was supported in part by the US DOE grant DE-FG05-85ER40226, the National Science Council of Taiwan under Grant Numbers 98-2112-M-001-014-MY3, 101-2112-M-001-005-MY3, and the National Center for Theoretical Sciences of Taiwan (NCTS). T.W.K. is grateful for the hospitality of the Physics Division of NCTS where this work was initiated. T.C.Y. is grateful for the hospitality of KITPC (Beijing, China) and IMSc (Chennai, India) where progress in this work was made.

Appendix

A.1 Discrete symmetry groups I′ and A₅

Fig. 1.

Extended Dynkin diagram and irreducible representations of I′.

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A.2 Higgs potential

The sector of the Higgs potential of the I′ model that depends only on |$H_1$|⁠, |$H_{1'}$|⁠, |$H_3$|⁠, and |$H_{3'}$| is given in the appendix of Ref. [57], which only involves T′ invariant terms and is identical to our form of the potential up to constraints due to residual I′ relations on the coupling constants. The work of Ref. [57] demonstrates how T′ is broken completely and how the light quark and lepton masses and mixings arise.

A similar analysis can be applied to the investigation of the SSB in our |$A_5$| model. Finally, we note that typically only an |$O(10^{-1})$| fine tuning of scalar quartic coupling constants is needed to maintain the stability of such patterns of SSB.

References