The Renormalizable B–L Model with D ₅ Discrete Symmetry for Lepton Masses and Mixings

Abstract

We construct a renormalizable B–L standard model (SM) extension with D₅ symmetry in which neutrino mass hierarchies and the tiny neutrino masses are generated at leading order by type I seesaw mechanism. The obtained physical parameters are well consistent with the global fit of neutrino oscillation data given in [1]. The model predicts an effective neutrino mass parameter of 〈m_ee〉 = 3.731 × 10^–3 eV for normal hierarchy (NH) and 〈m_ee〉 = 4.848 × 10^–2 eV for inverted hierarchy (IH) which are well consistent with the recent experimental limits on neutrinoless double beta decay.

Appendices

APPENDIX A

1.1 The Explicit Expressions of $\mathbb{K}$, ${{\mathbb{K}}_{{1,2}}}$, ${{\mathbb{N}}_{{1,2}}}$

$$\begin{gathered} \mathbb{K} = \frac{{{{c}_{D}} - {{d}_{D}}}}{{{{b}_{D}}}},\quad \frac{{{{\mathbb{K}}_{{1,2}}}}}{{{{b}_{D}}}} = \frac{{ - {{\alpha }_{1}} \mp \sqrt {{{\alpha }_{2}}} }}{{2{{\alpha }_{0}}}}, \\ {{\mathbb{N}}_{{1,2}}} = \frac{{ - {{\alpha }_{3}} \mp ({{c}_{D}} - {{d}_{D}})\sqrt {{{\alpha }_{2}}} }}{{2{{\alpha }_{0}}}}, \\ \end{gathered} $$

(A.1)

where

$$\begin{gathered} {{\alpha }_{0}} = a_{D}^{2}b_{R}^{2}({{c}_{D}} - {{d}_{D}}) + {{a}_{R}}\{ b_{D}^{2}[{{b}_{R}}{{c}_{D}} - ({{b}_{R}} + {{c}_{R}}){{d}_{D}}] \\ \, + ({{c}_{D}} - {{d}_{D}})[{{b}_{R}}(c_{D}^{2} + d_{D}^{2}) - {{c}_{D}}{{c}_{R}}{{d}_{D}}]\} , \\ \end{gathered} $$

$$\begin{gathered} {{\alpha }_{1}} = 2a_{D}^{2}b_{R}^{2} + {{a}_{R}}[(b_{D}^{2} + c_{D}^{2})(2{{b}_{R}} + {{c}_{R}}) \\ \, - 2{{c}_{D}}{{c}_{R}}{{d}_{D}} + (2{{b}_{R}} - {{c}_{R}})d_{D}^{2}], \\ \end{gathered} $$

$$\begin{gathered} {{\alpha }_{2}} = 4a_{D}^{4}b_{R}^{4} + 4a_{D}^{2}{{a}_{R}}b_{R}^{2}[b_{D}^{2}(2{{b}_{R}} + cR) - 2{{c}_{D}}{{c}_{R}}{{d}_{D}} \\ \, + 2{{b}_{R}}(c_{D}^{2} + d_{D}^{2})] + a_{R}^{2}(b_{D}^{2} + c_{D}^{2} + d_{D}^{2})[b_{D}^{2}{{(2{{b}_{R}} + {{c}_{R}})}^{2}} \\ \, - 8{{b}_{R}}{{c}_{D}}{{c}_{R}}{{d}_{D}} + 4b_{R}^{2}(c_{D}^{2} + d_{D}^{2}) + c_{R}^{2}(c_{D}^{2} + d_{D}^{2})], \\ \end{gathered} $$

$${{\alpha }_{3}} = {{a}_{R}}{{c}_{R}}[b_{D}^{2} + {{({{c}_{D}} - {{d}_{D}})}^{2}}]({{c}_{D}} + {{d}_{D}}).$$

(A.2)

APPENDIX B

1.1 Higgs Potential

The renormalizable potential invariant under all symmetries SU(3)_C $\otimes$ SU(2)_L $\otimes$ U(1)_Y $\otimes$ U(1)_B–L $\otimes$ $\underline{D}{}_5$ is given by:⁴

$$\begin{gathered} {{V}_{{{\text{total}}}}} = V(H) + V(\varphi ) + V(\phi ) + V(\chi ) + V(\eta ) \\ + \,V(H,\varphi ) + V(H,\phi ) + V(H,\chi ) + V(H,\eta ) \\ + \,V(\varphi ,\phi ) + V(\varphi ,\chi ) + V(\varphi ,\eta ) + V(\phi ,\chi ) \\ \, + V(\phi ,\eta ) + V(\chi ,\eta ), \\ \end{gathered} $$

(B.1)

where

$$V(H) = \mu _{H}^{2}{{H}^{\dag }}H + {{\lambda }^{H}}{{({{H}^{\dag }}H)}^{2}},$$

(B.2)

$$\begin{gathered} V(\varphi ) = \mu _{\varphi }^{2}{{\varphi }^{\dag }}\varphi + \lambda _{1}^{\varphi }{{({{\varphi }^{\dag }}\varphi )}_{\underline{1}{}_1}}{{({{\varphi }^{\dag }}\varphi )}_{\underline{1}{}_1}} \\ + \,\lambda _{2}^{\varphi }{{({{\varphi }^{\dag }}\varphi )}_{\underline{1}{}_2}}{{({{\varphi }^{\dag }}\varphi )}_{\underline{1}{}_2}} + \lambda _{3}^{\varphi }{{({{\varphi }^{\dag }}\varphi )}_{\underline{2}{}_2}}{{({{\varphi }^{\dag }}\varphi )}_{\underline{2}{}_2}}, \\ \end{gathered} $$

(B.3)

$$\begin{gathered} V(\phi ) = \mu _{\phi }^{2}{{\phi }^{\dag }}\phi + \lambda _{1}^{\phi }{{({{\phi }^{\dag }}\phi )}_{\underline{1}{}_1}}{{({{\phi }^{\dag }}\phi )}_{\underline{1}{}_1}} \\ + \,\lambda _{2}^{\phi }{{({{\phi }^{\dag }}\phi )}_{\underline{1}{}_2}}{{({{\phi }^{\dag }}\phi )}_{\underline{1}{}_2}} + \lambda _{3}^{\phi }{{({{\phi }^{\dag }}\phi )}_{\underline{2}{}_1}}{{({{\phi }^{\dag }}\phi )}_{\underline{2}{}_1}}, \\ \end{gathered} $$

(B.4)

$$V(\chi ) = V(H \to \chi ),\quad V(\eta ) = V(\varphi \to \eta ),$$

(B.5)

$$\begin{gathered} V(H,\varphi ) \\ = \,\lambda _{1}^{{H\varphi }}{{({{H}^{\dag }}H)}_{\underline{1}{}_1}}{{({{\varphi }^{\dag }}\varphi )}_{\underline{1}{}_1}}\, + \,\lambda _{2}^{{H\varphi }}{{({{H}^{\dag }}\varphi )}_{\underline{2}{}_1}}{{({{\varphi }^{\dag }}H)}_{\underline{2}{}_1}}, \\ \end{gathered} $$

(B.6)

$$\begin{gathered} V(H,\phi ) \\ = \lambda _{1}^{{H\phi }}{{({{H}^{\dag }}H)}_{\underline{1}{}_1}}{{({{\phi }^{\dag }}\phi )}_{\underline{1}{}_1}} + \lambda _{2}^{{H\phi }}{{({{H}^{\dag }}\phi )}_{\underline{2}{}_2}}{{({{\phi }^{\dag }}H)}_{\underline{2}{}_2}}, \\ \end{gathered} $$

(B.7)

$$\begin{gathered} V(H,\chi ) \\ = \lambda _{1}^{{H\chi }}{{({{H}^{\dag }}H)}_{\underline{1}{}_1}}{{({{\chi }^{\dag }}\chi )}_{\underline{1}{}_1}} + \lambda _{2}^{{H\chi }}{{({{H}^{\dag }}\chi )}_{\underline{1}{}_1}}{{({{\chi }^{\dag }}H)}_{\underline{1}{}_1}}, \\ \end{gathered} $$

(B.8)

$$V(H,\eta ) = V(H,\varphi \to \eta ),$$

(B.9)

$$\begin{gathered} V(\varphi ,\phi ) = \lambda _{1}^{{\varphi \phi }}{{({{\varphi }^{\dag }}\varphi )}_{\underline{1}{}_1}}{{({{\phi }^{\dag }}\phi )}_{\underline{1}{}_1}} + \lambda _{2}^{{\varphi \phi }}{{({{\varphi }^{\dag }}\varphi )}_{\underline{1}{}_2}}{{({{\phi }^{\dag }}\phi )}_{\underline{1}{}_2}} \\ \, + \lambda _{3}^{{\varphi \phi }}{{({{\varphi }^{\dag }}\phi )}_{\underline{2}{}_1}}{{({{\phi }^{\dag }}\varphi )}_{\underline{2}{}_1}} + \lambda _{4}^{{\varphi \phi }}{{({{\varphi }^{\dag }}\phi )}_{\underline{2}{}_2}}{{({{\phi }^{\dag }}\varphi )}_{\underline{2}{}_2}}, \\ \end{gathered} $$

(B.10)

$$V(\varphi ,\chi ) = \lambda _{1}^{{\varphi \chi }}{{({{\varphi }^{\dag }}\varphi )}_{\underline{1}{}_1}}{{({{\chi }^{\dag }}\chi )}_{\underline{1}{}_1}} + \lambda _{2}^{{\varphi \chi }}{{({{\varphi }^{\dag }}\chi )}_{\underline{2}{}_1}}{{({{\chi }^{\dag }}\varphi )}_{\underline{2}{}_1}},$$

(B.11)

$$\begin{gathered} V(\varphi ,\eta ) = \lambda _{1}^{{\varphi \eta }}{{({{\varphi }^{\dag }}\varphi )}_{\underline{1}{}_1}}{{({{\eta }^{\dag }}\eta )}_{\underline{1}{}_1}} + \lambda _{2}^{{\varphi \eta }}{{({{\varphi }^{\dag }}\varphi )}_{\underline{1}{}_2}}{{({{\eta }^{\dag }}\eta )}_{\underline{1}{}_2}} \\ \, + \lambda _{3}^{{\varphi \eta }}{{({{\varphi }^{\dag }}\varphi )}_{\underline{2}{}_2}}{{({{\eta }^{\dag }}\eta )}_{\underline{2}{}_2}} + \lambda _{4}^{{\varphi \eta }}{{({{\varphi }^{\dag }}\eta )}_{\underline{1}{}_1}}{{({{\eta }^{\dag }}\varphi )}_{\underline{1}{}_1}} \\ \, + \lambda _{5}^{{\varphi \eta }}{{({{\varphi }^{\dag }}\eta )}_{\underline{1}{}_2}}{{({{\eta }^{\dag }}\varphi )}_{\underline{1}{}_2}} + \lambda _{6}^{{\varphi \eta }}{{({{\varphi }^{\dag }}\eta )}_{\underline{2}{}_2}}{{({{\eta }^{\dag }}\varphi )}_{\underline{2}{}_2}}, \\ \end{gathered} $$

(B.12)

$$V(\phi ,\chi ) = \lambda _{1}^{{\phi \chi }}{{({{\phi }^{\dag }}\phi )}_{\underline{1}{}_1}}{{({{\chi }^{\dag }}\chi )}_{\underline{1}{}_1}} + \lambda _{2}^{{\phi \chi }}{{({{\phi }^{\dag }}\chi )}_{\underline{2}{}_2}}{{({{\chi }^{\dag }}\phi )}_{\underline{2}{}_2}},$$

(B.13)

$$\begin{gathered} V(\phi ,\eta ) = \lambda _{1}^{{\phi \eta }}{{({{\phi }^{\dag }}\phi )}_{\underline{1}{}_1}}{{({{\eta }^{\dag }}\eta )}_{\underline{1}{}_1}} + \lambda _{2}^{{\phi \eta }}{{({{\phi }^{\dag }}\phi )}_{\underline{1}{}_2}}{{({{\eta }^{\dag }}\eta )}_{\underline{1}{}_2}} \\ \, + \lambda _{3}^{{\phi \eta }}{{({{\phi }^{\dag }}\eta )}_{\underline{2}{}_1}}{{({{\eta }^{\dag }}\phi )}_{\underline{2}{}_1}} + \lambda _{4}^{{\phi \eta }}{{({{\phi }^{\dag }}\eta )}_{\underline{2}{}_2}}{{({{\eta }^{\dag }}\phi )}_{\underline{2}{}_2}}, \\ V(\chi ,\eta ) = V(H \to \chi ,\varphi \to \eta ). \\ \end{gathered} $$

(B.14)

The scalars fields H, φ, ϕ, χ, and η with their VEVs aligned in Eq. (3) is a solution from the minimization condition of V_total. To see this, in the system of minimization equations, let us put ${{{v}}_{{{{\phi }_{1}}}}}$ = ${{{v}}_{{{{\phi }_{2}}}}}$ = ${{{v}}_{\phi }}$, ${{{v}}_{{{{\eta }_{1}}}}}$ = 0, ${{{v}}_{{{{\eta }_{1}}}}}$ = ${{{v}}_{\eta }}$ and ${v}{\text{*}}$ = ${v}$, ${v}_{\phi }^{*}$ = ${{{v}}_{\phi }}$, ${v}_{\chi }^{*}$ = ${{{v}}_{\chi }}$, ${v}_{\eta }^{*}$ = ${{{v}}_{\eta }}$, which reduces to

$$\begin{gathered} \mu _{H}^{2} + (\lambda _{1}^{{H\varphi }} + \lambda _{2}^{{H\varphi }})({v}_{1}^{*}{{{v}}_{2}}\, + \,{v}_{2}^{*}{{{v}}_{1}})\, + \,(\lambda _{1}^{{H\chi }} + \lambda _{2}^{{H\chi }}){v}_{\chi }^{2} \\ \, + 2[(\lambda _{1}^{{H\eta }}\, + \,\lambda _{2}^{{H\eta }}){v}_{\eta }^{2} + {{\lambda }^{H}}{{{v}}^{2}}\, + \,(\lambda _{1}^{{H\phi }} + \lambda _{2}^{{H\phi }}){v}_{\phi }^{2}] = 0, \\ \end{gathered} $$

(B.15)

$$\begin{gathered} \mu _{\varphi }^{2}{v}_{2}^{*} + 2\lambda _{3}^{\varphi }{v}_{1}^{*}{v}_{2}^{{}}{v}_{2}^{*} + 2\lambda _{2}^{\varphi }{v}_{2}^{*}({v}_{1}^{{}}{v}_{2}^{*} - {v}_{1}^{*}{v}_{2}^{{}}) \\ \, + 2\lambda _{1}^{\varphi }{v}_{2}^{*}({v}_{1}^{{}}{v}_{2}^{*} + {v}_{1}^{*}{v}_{2}^{{}}) + [(\lambda _{3}^{{\varphi \eta }} + \lambda _{4}^{{\varphi \eta }} - \lambda _{5}^{{\varphi \eta }}){v}_{1}^{*} \\ \, + (2\lambda _{1}^{{\varphi \eta }} + \lambda _{4}^{{\varphi \eta }} + \lambda _{5}^{{\varphi \eta }} + \lambda _{6}^{{\varphi \eta }}){v}_{2}^{*}]{v}_{\eta }^{2} \\ \, + (\lambda _{1}^{\varphi } + \lambda _{2}^{\varphi }){v}_{2}^{*}{v}_{\chi }^{2} + (\lambda _{1}^{{H\eta }} + \lambda _{2}^{{H\eta }}){v}_{2}^{*}{{{v}}^{2}} \\ \, + [\lambda _{3}^{{\varphi \phi }}{v}_{1}^{*} + (2\lambda _{1}^{{\varphi \phi }} + \lambda _{4}^{{\varphi \phi }}){v}_{2}^{*}]{v}_{\phi }^{2} = 0, \\ \end{gathered} $$

(B.16)

$$\begin{gathered} \mu _{\varphi }^{2}{v}_{1}^{*} + 2\lambda _{3}^{\varphi }{v}_{1}^{{}}{v}_{1}^{*}{v}_{2}^{*} + 2\lambda _{2}^{\varphi }{v}_{1}^{*}({v}_{1}^{*}{{{v}}_{2}} - {{{v}}_{1}}{v}_{2}^{*}) \\ \, + 2\lambda _{1}^{\varphi }{v}_{1}^{*}({v}_{1}^{*}{{{v}}_{2}} + {{{v}}_{1}}{v}_{2}^{*}) + [(\lambda _{3}^{{\varphi \eta }} + \lambda _{4}^{{\varphi \eta }} - \lambda _{5}^{{\varphi \eta }}){v}_{2}^{*} \\ \, + (2\lambda _{1}^{{\varphi \eta }} + \lambda _{4}^{{\varphi \eta }} + \lambda _{5}^{{\varphi \eta }} + \lambda _{6}^{{\varphi \eta }}){v}_{1}^{*}]{v}_{\eta }^{2} \\ \, + (\lambda _{1}^{\varphi } + \lambda _{2}^{\varphi }){v}_{1}^{*}{v}_{\chi }^{2} + (\lambda _{1}^{{H\eta }} + \lambda _{2}^{{H\eta }}){v}_{1}^{*}{{{v}}^{2}} \\ \, + [\lambda _{3}^{{\varphi \phi }}{v}_{2}^{*} + (2_{1}^{{\varphi \phi }} + \lambda _{4}^{{\varphi \phi }}){v}_{1}^{*}]{v}_{\phi }^{2} = 0, \\ \end{gathered} $$

(B.17)

$$\begin{gathered} 2[(\lambda _{1}^{{\phi \chi }} + \lambda _{2}^{{\phi \chi }}){v}_{\chi }^{2} + (2\lambda _{1}^{{\phi \eta }} + \lambda _{3}^{{\phi \eta }} + \lambda _{4}^{{\phi \eta }}){v}_{\eta }^{2} \\ \, + (\lambda _{1}^{{H\phi }} + \lambda _{2}^{{H\phi }}){{{v}}^{2}} + (4\lambda _{1}^{\phi } + 2\lambda _{3}^{\phi }){v}_{\phi }^{2}] + 2\mu _{\phi }^{2} \\ + \,(2\lambda _{1}^{\varphi }\, + \,\lambda _{4}^{\varphi })({v}_{1}^{*}{{{v}}_{2}}\, + \,{v}_{2}^{*}{{{v}}_{1}})\, + \,\lambda _{3}^{\varphi }({\text{|}}{{{v}}_{1}}{{{\text{|}}}^{2}}\, + \,{\text{|}}{{{v}}_{2}}{{{\text{|}}}^{2}})\, = \,0, \\ \end{gathered} $$

(B.18)

$$\begin{gathered} \mu _{\chi }^{2} + (\lambda _{1}^{{\varphi \chi }} + \lambda _{2}^{{\varphi \chi }})({v}_{1}^{*}{{{v}}_{2}} + {v}_{2}^{*}{{{v}}_{1}}) \\ \, + 2{{\lambda }^{\chi }}{v}_{\chi }^{2} + 2(\lambda _{1}^{{\chi \eta }} + \lambda _{2}^{{\chi \eta }}){v}_{\eta }^{2} \\ \, + (\lambda _{1}^{{H\chi }} + \lambda _{2}^{{H\chi }}){{{v}}^{2}} + 2(\lambda _{1}^{{\phi \chi }} + \lambda _{2}^{{\phi \chi }}){v}_{\phi }^{2} = 0, \\ \end{gathered} $$

(B.19)

$$\begin{gathered} 2\mu _{\eta }^{2} + (\lambda _{3}^{{\varphi \eta }} + \lambda _{4}^{{\varphi \eta }} - \lambda _{5}^{{\varphi \eta }}){\text{|}}{{{v}}_{1}}{{{\text{|}}}^{2}} \\ \, + (2\lambda _{1}^{{\varphi \eta }} + \lambda _{4}^{{\varphi \eta }} + \lambda _{5}^{{\varphi \eta }} + \lambda _{6}^{{\varphi \eta }}){v}_{1}^{*}{{{v}}_{2}} \\ \, + 2[(\lambda _{1}^{{\chi \eta }} + \lambda _{2}^{{\chi \eta }}){v}_{\chi }^{2} + 2(2\lambda _{1}^{\eta } + \lambda _{3}^{\eta }){v}_{\eta }^{2} \\ \, + (\lambda _{1}^{{H\eta }} + \lambda _{2}^{{H\eta }}){{{v}}^{2}} + (2\lambda _{1}^{{\phi \eta }} + \lambda _{3}^{{\phi \eta }} + \lambda _{4}^{{\phi \eta }}){v}_{\phi }^{2}] \\ \, + (2\lambda _{1}^{{\varphi \eta }} + \lambda _{4}^{{\varphi \eta }} + \lambda _{5}^{{\varphi \eta }} + \lambda _{6}^{{\varphi \eta }}){v}_{2}^{*}{{{v}}_{1}} \\ \, + (\lambda _{3}^{{\varphi \eta }} + \lambda _{4}^{{\varphi \eta }} - \lambda _{5}^{{\varphi \eta }}{\text{)|}}{{{v}}_{2}}{{{\text{|}}}^{2}} = 0. \\ \end{gathered} $$

(B.20)

Because the number of equations for the potential minimization (B.15)–(B.20) are less than the number of Higgs potential parameters, so this system of equation always give the solution (${v}$, ${{{v}}_{1}}$, ${{{v}}_{2}}$, ${{{v}}_{\phi }}$, ${{{v}}_{\chi }}$, ${{{v}}_{\eta }}$) as chosen in Eq. (3). It is noted that the alignments in Eq. (3) is only one solution to have the desirable results. There are some other solutions but that the physics results are different. For instance, the solution with the alignment 〈η〉 = (〈η₁〉, 〈η₂〉) (and the aligments of the other scalars are given in Eq. (3)) provides an additional contribution to the element (22) of the matrix ${{\mathbb{M}}_{R}}$ in Eq. (15) and model thus contains one more parameter; the solution with the alignment 〈η₁〉 = 〈η₂〉 ≠ 0 provides the analytical expressions for the neutrino masses in simpler form, however, it cannot accommodate the neutrino mixing data in Eq. (1), etc. Therefore, the solution in Eq. (3) is preferred for this work.