Predictive Pati-Salam theory of fermion masses and mixing

TL;DR

This paper introduces a Pati-Salam extension with a flavor symmetry that predicts specific neutrino oscillation patterns, CP violation, and links to potential collider signals, advancing the understanding of fermion masses.

Contribution

It develops a novel Pati-Salam model with a $ riangle(27)$ flavor symmetry that provides predictive neutrino oscillation patterns and connects low-scale seesaw mechanisms to collider phenomenology.

Findings

Normal hierarchy predicts higher octant atmospheric angle and CP violation.

Inverted hierarchy favors lower octant and possible CP conservation.

Lower bound on neutrinoless double beta decay rate.

Abstract

We propose a Pati-Salam extension of the standard model incorporating a flavor symmetry based on the $\Delta \left( 27\right)$ group. The theory realizes a realistic Froggatt-Nielsen picture of quark mixing and a predictive pattern of neutrino oscillations. We find that, for normal neutrino mass ordering, the atmospheric angle must lie in the higher octant, CP must be violated in oscillations, and there is a lower bound for the $0\nu\beta\beta$ decay rate. For the case of inverted mass ordering, we find that the lower atmospheric octant is preferred, and that CP can be conserved in oscillations. Neutrino masses arise from a low-scale seesaw mechanism, whose messengers can be produced by a $Z^{\prime }$ portal at the LHC.

Tables

Table 1. Table 1: Particle content and transformation properties under the gauge symmetry.

Field

${\overline{F}}_{iL}$

$F_{iR}$

${\overline{\Psi }}_{iL}$

${\Psi }_{iR}$

$S_i$

${\chi }_L$

${\chi }_R$

${\Phi }_j$

$\Sigma$

${\varphi }_k$

${\phi }_l$

$\sigma$

${\rho }_k$

${\eta }_k$

${\tau }_k$

${\xi }_k$

$SU{\left(4\right)}_C$

$\overline{\mathrm{𝟒}}$

$\mathrm{𝟒}$

$\overline{\mathrm{𝟒}}$

$\mathrm{𝟒}$

$\mathrm{𝟏}$

$\mathrm{𝟒}$

$\mathrm{𝟒}$

$\mathrm{𝟏}$

$\mathrm{𝟏𝟓}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$SU{\left(2\right)}_L$

$\mathrm{𝟐}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟐}$

$\mathrm{𝟏}$

$\mathrm{𝟐}$

$\mathrm{𝟏}$

$\mathrm{𝟐}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$SU{\left(2\right)}_R$

$\mathrm{𝟏}$

$\mathrm{𝟐}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟐}$

$\mathrm{𝟐}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟐}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

$\mathrm{𝟏}$

Table 2. Table 2: Transformation properties of the leptons and quarks under the flavor symmetry Δ ​ ( 27 ) ⊗ Z 4 ⊗ Z 4 ′ tensor-product Δ 27 subscript 𝑍 4 superscript subscript 𝑍 4 ′ \Delta(27)\otimes Z_{4}\otimes Z_{4}^{\prime} .

Field	${\overline{F}}_{1L}$	${\overline{F}}_{2L}$	${\overline{F}}_{3L}$	$F_{1R}$	$F_{2R}$	$F_{3R}$	${\overline{\Psi }}_{1L}$	${\overline{\Psi }}_{2L}$	${\overline{\Psi }}_{3L}$	${\Psi }_{1R}$	${\Psi }_{2R}$	${\Psi }_{3R}$	$S$
$\Delta \left(27\right)$	${\mathrm{𝟏}}_{\mathrm{𝟎},\mathrm{𝟎}}$	${\mathrm{𝟏}}_{\mathrm{𝟐},\mathrm{𝟎}}$	${\mathrm{𝟏}}_{\mathrm{𝟏},\mathrm{𝟎}}$	${\mathrm{𝟏}}_{\mathrm{𝟎},\mathrm{𝟎}}$	${\mathrm{𝟏}}_{\mathrm{𝟏},\mathrm{𝟎}}$	${\mathrm{𝟏}}_{\mathrm{𝟐},\mathrm{𝟎}}$	${\mathrm{𝟏}}_{\mathrm{𝟎},\mathrm{𝟎}}$	${\mathrm{𝟏}}_{\mathrm{𝟐},\mathrm{𝟎}}$	${\mathrm{𝟏}}_{\mathrm{𝟏},\mathrm{𝟎}}$	${\mathrm{𝟏}}_{\mathrm{𝟎},\mathrm{𝟎}}$	${\mathrm{𝟏}}_{\mathrm{𝟏},\mathrm{𝟎}}$	${\mathrm{𝟏}}_{\mathrm{𝟐},\mathrm{𝟎}}$	$\mathrm{𝟑}$
$Z_4$	$1$	$i$	$1$	$1$	$i$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$i$
$Z_{16}$	$i$	$i^{\frac{1}{2}}$	$1$	$i$	$i^{\frac{1}{2}}$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$

Table 3. Table 3: Transformation properties of the scalars under the flavor symmetry Δ ​ ( 27 ) ⊗ Z 4 ⊗ Z 4 ′ tensor-product Δ 27 subscript 𝑍 4 superscript subscript 𝑍 4 ′ \Delta(27)\otimes Z_{4}\otimes Z_{4}^{\prime} .

Field

${\chi }_L$

${\chi }_R$

${\Phi }_1$

${\Phi }_2$

$\Sigma$

${\varphi }_1$

${\varphi }_2$

${\varphi }_3$

${\phi }_1$

${\phi }_2$

${\phi }_3$

${\phi }_4$

${\phi }_5$

$\sigma$

$\rho$

$\eta$

$\tau$

$\xi$

$\Delta \left(27\right)$

${\mathrm{𝟏}}_{\mathrm{𝟎},\mathrm{𝟎}}$

${\mathrm{𝟏}}_{\mathrm{𝟎},\mathrm{𝟎}}$

${\mathrm{𝟏}}_{\mathrm{𝟎},\mathrm{𝟎}}$

${\mathrm{𝟏}}_{\mathrm{𝟎},\mathrm{𝟎}}$

${\mathrm{𝟏}}_{\mathrm{𝟎},\mathrm{𝟎}}$

${\mathrm{𝟏}}_{\mathrm{𝟎},\mathrm{𝟎}}$

${\mathrm{𝟏}}_{\mathrm{𝟎},\mathrm{𝟎}}$

${\mathrm{𝟏}}_{\mathrm{𝟎},\mathrm{𝟎}}$

${\mathrm{𝟏}}_{\mathrm{𝟎},\mathrm{𝟎}}$

${\mathrm{𝟏}}_{\mathrm{𝟎},\mathrm{𝟎}}$

${\mathrm{𝟏}}_{\mathrm{𝟎},\mathrm{𝟎}}$

${\mathrm{𝟏}}_{\mathrm{𝟐},\mathrm{𝟎}}$

${\mathrm{𝟏}}_{\mathrm{𝟐},\mathrm{𝟎}}$

${\mathrm{𝟏}}_{\mathrm{𝟎},\mathrm{𝟎}}$

$\overline{\mathrm{𝟑}}$

$\overline{\mathrm{𝟑}}$

$\overline{\mathrm{𝟑}}$

$\overline{\mathrm{𝟑}}$

$Z_4$

$-i$

$i$

$1$

$-1$

$1$

$1$

$-i$

$1$

$1$

$-i$

$1$

$-i$

$1$

$1$

$1$

$-i$

$1$

$-1$

$Z_{16}$

$1$

$1$

$1$

$1$

$i^{\frac{3}{2}}$

$i^{-\frac{1}{2}}$

$i^{-\frac{1}{4}}$

$1$

$i^{-\frac{1}{2}}$

$i^{-\frac{1}{4}}$

$1$

$i^{-\frac{1}{4}}$

$i^{\frac{1}{4}}$

$i^{-\frac{1}{4}}$

$i$

$i^{\frac{1}{2}}$

$1$

$1$

Table 4. Table 4: Model and experimental values of the quark masses and CKM parameters.

Observable	Model value	Experimental value
$m_u\left(\mathrm{MeV}\right)$	$1.45$	${1.45}_{-0.45}^{+0.56}$
$m_c\left(\mathrm{MeV}\right)$	$635$	$635\pm 86$
$m_t\left(\mathrm{GeV}\right)$	$172$	$172.1\pm 0.6\pm 0.9$
$m_d\left(\mathrm{MeV}\right)$	$2.9$	${2.9}_{-0.4}^{+0.5}$
$m_s\left(\mathrm{MeV}\right)$	$57.7$	${57.7}_{-15.7}^{+16.8}$
$m_b\left(\mathrm{GeV}\right)$	$2.82$	${2.82}_{-0.04}^{+0.09}$
$\sin {\theta }_{12}$	$0.225$	$0.225$
$\sin {\theta }_{23}$	$0.0411$	$0.0411$
$\sin {\theta }_{13}$	$0.00357$	$0.00357$
$\delta$	$1.236$	$1.236$

Table 5. Table 5: Characters of Δ ​ ( 27 ) Δ 27 \Delta(27)

	h	${\chi }_{1_{(r,s)}}$	${\chi }_{3_{[0,1]}}$	${\chi }_{3_{[0,2]}}$
$1C_1$	1	1	3	3
$1C_1^{\left(1\right)}$	1	1	$3{\omega }^2$	$3\omega$
$1C_1^{\left(2\right)}$	1	1	$3\omega$	$3{\omega }^2$
$3C_1^{(0,1)}$	$3$	${\omega }^s$	$0$	$0$
$3C_1^{(0,2)}$	$3$	${\omega }^{2s}$	$0$	$0$
$C_3^{(1,p)}$	$3$	${\omega }^{r+sp}$	0	0
$C_3^{(2,p)}$	$3$	${\omega }^{2r+sp}$	0	0

Table 6. Table 6: The singlet multiplications of the group Δ ​ ( 27 ) Δ 27 \Delta(27) .

Singlets	${\mathrm{𝟏}}_{01}$	${\mathrm{𝟏}}_{02}$	${\mathrm{𝟏}}_{10}$	${\mathrm{𝟏}}_{11}$	${\mathrm{𝟏}}_{12}$	${\mathrm{𝟏}}_{20}$	${\mathrm{𝟏}}_{21}$	${\mathrm{𝟏}}_{22}$
${\mathrm{𝟏}}_{01}$	${\mathrm{𝟏}}_{02}$	${\mathrm{𝟏}}_{00}$	${\mathrm{𝟏}}_{11}$	${\mathrm{𝟏}}_{12}$	${\mathrm{𝟏}}_{10}$	${\mathrm{𝟏}}_{21}$	${\mathrm{𝟏}}_{22}$	${\mathrm{𝟏}}_{20}$
${\mathrm{𝟏}}_{02}$	${\mathrm{𝟏}}_{00}$	${\mathrm{𝟏}}_{01}$	${\mathrm{𝟏}}_{12}$	${\mathrm{𝟏}}_{10}$	${\mathrm{𝟏}}_{11}$	${\mathrm{𝟏}}_{22}$	${\mathrm{𝟏}}_{20}$	${\mathrm{𝟏}}_{21}$
${\mathrm{𝟏}}_{10}$	${\mathrm{𝟏}}_{11}$	${\mathrm{𝟏}}_{12}$	${\mathrm{𝟏}}_{20}$	${\mathrm{𝟏}}_{21}$	${\mathrm{𝟏}}_{22}$	${\mathrm{𝟏}}_{00}$	${\mathrm{𝟏}}_{01}$	${\mathrm{𝟏}}_{02}$
${\mathrm{𝟏}}_{11}$	${\mathrm{𝟏}}_{12}$	${\mathrm{𝟏}}_{10}$	${\mathrm{𝟏}}_{21}$	${\mathrm{𝟏}}_{22}$	${\mathrm{𝟏}}_{20}$	${\mathrm{𝟏}}_{01}$	${\mathrm{𝟏}}_{02}$	${\mathrm{𝟏}}_{00}$
${\mathrm{𝟏}}_{12}$	${\mathrm{𝟏}}_{10}$	${\mathrm{𝟏}}_{11}$	${\mathrm{𝟏}}_{22}$	${\mathrm{𝟏}}_{20}$	${\mathrm{𝟏}}_{21}$	${\mathrm{𝟏}}_{02}$	${\mathrm{𝟏}}_{00}$	${\mathrm{𝟏}}_{01}$
${\mathrm{𝟏}}_{20}$	${\mathrm{𝟏}}_{21}$	${\mathrm{𝟏}}_{22}$	${\mathrm{𝟏}}_{00}$	${\mathrm{𝟏}}_{01}$	${\mathrm{𝟏}}_{02}$	${\mathrm{𝟏}}_{10}$	${\mathrm{𝟏}}_{11}$	${\mathrm{𝟏}}_{12}$
${\mathrm{𝟏}}_{21}$	${\mathrm{𝟏}}_{22}$	${\mathrm{𝟏}}_{20}$	${\mathrm{𝟏}}_{01}$	${\mathrm{𝟏}}_{02}$	${\mathrm{𝟏}}_{00}$	${\mathrm{𝟏}}_{11}$	${\mathrm{𝟏}}_{12}$	${\mathrm{𝟏}}_{10}$
${\mathrm{𝟏}}_{22}$	${\mathrm{𝟏}}_{20}$	${\mathrm{𝟏}}_{21}$	${\mathrm{𝟏}}_{02}$	${\mathrm{𝟏}}_{00}$	${\mathrm{𝟏}}_{01}$	${\mathrm{𝟏}}_{12}$	${\mathrm{𝟏}}_{10}$	${\mathrm{𝟏}}_{11}$

Equations

$${\overline{F}}_{iL}$$

$${\overline{\Psi }}_{iL}$$

$$\overline{F}_{iL}=\left(\begin{array}[]{cccc}\overline{u}_{iL}&\overline{u}_{iL}&\overline{u}_{iL}&\overline{\nu}_{iL}\\ \overline{d}_{iL}&\overline{d}_{iL}&\overline{d}_{iL}&\overline{l}_{iL}\end{array}\right)^{T},\hskip 42.67912ptF_{iR}=\left(\begin{array}[]{cccc}u_{iR}&u_{iR}&u_{iR}&\nu_{iR}\\ d_{iR}&d_{iR}&d_{iR}&l_{iR}\end{array}\right).$$

$${\overline{F}}_{1L}$$

$$F_{1R}$$

$${\overline{\Psi }}_{1L}$$

$${\Psi }_{1R}$$

$$S$$

$${\chi }_R$$

$${\varphi }_k$$

$${\eta }_k$$

$$\left\langle\chi_{L,R}\right\rangle=\left(\begin{array}[]{cccc}0&0&0&v_{L,R}\\ 0&0&0&0\end{array}\right),\hskip 42.67912pt\left\langle\Phi_{j}\right\rangle=\left(\begin{array}[]{cc}v_{1}^{\left(j\right)}&0\\ 0&v_{2}^{\left(j\right)}\end{array}\right),\hskip 42.67912pt\left\langle\Sigma\right\rangle=v_{\Sigma}T^{15},$$

$${\chi }_R$$

$$\Sigma$$

$${\phi }_1$$

$${\phi }_5$$

$$\tau$$

$$\rho$$

$$-{\mathcal{L}}_Y$$

$$\mathcal{G}=SU(4{)}_C\otimes SU{\left(2\right)}_L\otimes SU{\left(2\right)}_R\otimes \Delta \left(27\right)\otimes Z_4\otimes Z_{16}$$

$$⟨\Sigma ⟩\sim {\Lambda }_{PS}$$

$$\Downarrow$$

$$SU(3{)}_C\otimes SU{\left(2\right)}_L\otimes SU{\left(2\right)}_R\otimes U{\left(1\right)}_{B-L}$$

$$\displaystyle\langle\chi_{R}\rangle\sim\ v_{R}\ \$$

$$\Downarrow$$

$$SU(3{)}_C\otimes SU{\left(2\right)}_L\otimes U{\left(1\right)}_Y$$

$$\displaystyle\langle\Phi_{{1,2}}\rangle\sim\ v\ \ \ \$$

$$\Downarrow$$

$$SU(3{)}_C\otimes U{\left(1\right)}_Q.$$

$$v_L\ll v_{\xi }\ll v_1^{\left(j\right)}\sim v_{{\varphi }_i}\sim v_{{\phi }_l}\sim v\ll v_R\ll v_{\rho }=v_{\eta }=v_{\tau }\sim {\Lambda }_{PS}\sim v_{\sigma }=\lambda \Lambda ,$$

$$⟨\rho ⟩$$

$$⟨\xi ⟩$$

$$U_1^{(a)}$$

$$U_1^{(b)}$$

$$M_{U}=\left(\begin{array}[]{ccc}y_{1}\lambda^{8}&0&0\\ 0&y_{2}\lambda^{4}&0\\ 0&0&y_{3}\end{array}\right)\allowbreak\frac{v}{\sqrt{2}},$$

$$\left(\begin{array}[]{cc}\overline{d}_{iL}&\overline{D}_{iL}\end{array}\right)\left(\begin{array}[]{cc}0_{3\times 3}&M_{a}\\ M_{b}&M_{D}\end{array}\right)\left(\begin{array}[]{c}d_{jR}\\ D_{jR}\end{array}\right)\,,\qquad\left(\begin{array}[]{cc}\overline{l}_{iL}&\overline{L}_{iL}\end{array}\right)\left(\begin{array}[]{cc}0_{3\times 3}&M_{a}\\ M_{b}&M_{l}\end{array}\right)\left(\begin{array}[]{c}l_{jR}\\ L_{jR}\end{array}\right),$$

$$M_a$$

$$M_D$$

$$M_l$$

$$M_{f}^{\left(1\right)}=M_{a}\left(M_{f}\right)^{-1}M_{b}=\left(\begin{array}[]{ccc}a_{1f}\lambda^{7}&a_{4f}\lambda^{6}&a_{5f}\lambda^{6}\\ 0&a_{2f}\lambda^{5}&a_{6f}\lambda^{5}\\ 0&0&a_{3f}\lambda^{3}\end{array}\right)\frac{v}{\sqrt{2}},\hskip 28.45274ptf=D,l.$$

$$\begin{array}[]{c}y_{1}\simeq 1.269\,,\hskip 28.45274pty_{2}\simeq 1.424\,,\hskip 28.45274pty_{3}\simeq 0.989\,,\\ a_{1D}\simeq 0.585\,,\hskip 28.45274pta_{2D}\simeq 0.560\,,\hskip 28.45274pta_{3D}\simeq 1.421\,,\\ a_{4D}\simeq 0.573\,,\hskip 28.45274pt\left|a_{5D}\right|\simeq 0.446\,,\hskip 28.45274pt\arg(a_{5D})\simeq-1.906\,,\hskip 28.45274pta_{6D}\simeq 1.153\,.\end{array}$$

$$-\mathcal{L}_{mass}^{\left(\nu\right)}=\frac{1}{2}\left(\begin{array}[]{ccc}\overline{\nu_{L}^{C}}&\overline{\nu_{R}}&\overline{S}\end{array}\right)M_{\nu}\left(\begin{array}[]{c}\nu_{L}\\ \nu_{R}^{C}\\ S^{C}\end{array}\right)+H.c,$$

Full text

1 Introduction

Apart from the discovery of neutrino oscillations Kajita:2016cak ; McDonald:2016ixn , no other laboratory evidence for physics beyond the standard model has been so far unambiguously confirmed. Both the origin of neutrino mass itself, as well as the understanding of the mixing pattern, require an explanation from first principles. Moreover, there is a variety of other motivations for having beyond the standard model physics Valle:2015pba . One of these is the pursuit of a dynamical explanation for the origin of parity violation in the weak interaction, whose basic V-A nature is put in by hand in the formulation of the standard model. With this in mind here we propose a flavored Ishimori:2010au ; Morisi:2012fg ; King:2013eh Pati-Salam pati:1974yy extension of the standard model, addressing both the dynamical origin of the V-A nature of the weak force, as well as the related origin of neutrino mass. In addition, as we will see, the model can shed light upon the flavor problem and make predictions. The main features of our model include:

• •

adequate implementation of $\Delta\left(27\right)$ flavor symmetry in the Pati-Salam framework and symmetry breaking

• •

consistent low-scale left-right symmetric seesaw mechanism for neutrinos Mohapatra:1986bd ; Akhmedov:1995ip ; Akhmedov:1995vm ; Malinsky:2005bi ; Perez:2013osa

• •

predictive pattern of neutrino mixing summarized in Figs. 1 and 2

• •

realistic pattern of quark mixing, yielding a Froggatt-Nielsen Froggatt:1978nt -like picture of the CKM matrix

• •

lower bound for the $\rm 0\nu\beta\beta$ decay rate in Fig. 3

We note also that the model has a low-scale $Z^{\prime}$ portal through which the TeV scale messengers $S_{a}$ can be pair-produced in Drell-Yan collisions at the LHC Deppisch:2013cya ; AguilarSaavedra:2012fu ; Das:2012ii . In addition, our model realizes a universal seesaw mechanism Davidson:1987mh for the down type quarks as well as the charged leptons, mediated by TeV scale exotic fermions. The latter should potentially lead to other phenomenological effects in the quark sector as well as lepton flavour violation effects.

2 The model

The model under consideration is based on the Pati-Salam gauge symmetry $SU(4)_{C}\otimes SU\left(2\right)_{L}\otimes SU\left(2\right)_{R}$, supplemented by the $\Delta\left(27\right)\otimes Z_{4}\otimes Z_{16}$ discrete family symmetry group. The fermion transformation properties under the Pati-Salam group are

[TABLE]

where the subscript refers to fermion families. More explicitly, the standard model fermions are written in component form as

[TABLE]

Notice that we have extended the fermion sector of the original Pati-Salam model pati:1974yy by introducing three fermion singlets $S_{i}$, in order to implement inverse and/or linear seesaw mechanisms for the generation of light active neutrino masses Mohapatra:1986bd ; Akhmedov:1995ip ; Akhmedov:1995vm ; Malinsky:2005bi ; Perez:2013osa . In addition, we have introduced vector-like fermions $\overline{\Psi}_{iL}$ and $\Psi_{iR}$ so as to generate the standard model down-type quark and charged lepton masses, from a universal seesaw mechanism.

The particle content and gauge symmetry assignments are summarized in table 1.

The fermion assignments under the flavor symmetry group $\Delta\left(27\right)\otimes Z_{4}\otimes Z_{16}$ are:

[TABLE]

where the numbers in boldface stand for the dimensions of the $\Delta\left(27\right)$ irreducible representations, and we have defined $S\equiv\left(S_{1},S_{2},S_{3}\right)$. Notice that the three Pati-Salam singlet fermions are grouped into a $\Delta\left(27\right)$ triplet $S$, whereas the remaining fermions are assigned to singlet representations of $\Delta\left(27\right)$. These fermion flavor symmetry transformation properties are summarized in table 2. A brief summary of the properties of $\Delta\left(27\right)$ group and its irreducible representations is given in Appendix A.

We now turn to the main features of the scalar sector. The Higgs fields of the model are assumed to transform under the Pati-Salam gauge group as

[TABLE]

and the scalars $\chi_{L,R}$, $\Phi_{k}$ and $\Sigma$ develop vacuum expectation values (VEVs) of the form

[TABLE]

with $T^{15}=\frac{1}{2\sqrt{6}}\mathrm{diag}\left(1,1,1,-3\right)$.

The transformation properties of the scalar fields under the $\Delta\left(27\right)\otimes Z_{4}\otimes Z_{16}$ discrete group are given as follows:

[TABLE]

where we have set

[TABLE]

These flavor symmetry scalar transformation properties are summarized in table 3.

With the above particle content and transformation properties the following relevant Yukawa terms arise:

[TABLE]

where $r$, $s$ are $SU\left(4\right)$ indices and $y_{i}$, $\alpha_{i}$, $\beta_{i}$, $\kappa_{i}$, $a_{i}$ ($i=1,2,3$), $\gamma_{m}$ ($m=1,2,\cdots,6$) and $\lambda_{j}$ ($j=1,2$) are $\mathcal{O}(1)$ dimensionless couplings. For an explanation of the $\Delta(27)$ notation used in the $\alpha,\beta$ and $\lambda$-terms, see Appendix A. It is noteworthy that the lightest of the physical neutral scalar states of $\left(\Phi_{j}\right)_{11}$, $\left(\Phi_{j}\right)_{22}$, $\phi_{i}$ should be interpreted as the SM-like 125 GeV Higgs recently found at the LHC. Furthermore, our model at low energies corresponds to a seven Higgs doublet model with five scalar singlets (these scalar singlets come from $\varphi_{l}$). As we will show in section 4, the top quark mass only arises from $\left(\Phi_{1}\right)_{11}$. Consequently, the dominant contribution to the SM-like 125 GeV Higgs mainly arises from $\left(\Phi_{1}\right)_{11}$. We note also that the scalar potential of our model has many free parameters, so that we can assume the remaining scalars to be heavy and outside the LHC reach. Moreover, one can suppress the loop effects of the heavy scalars contributing to precision observables, by making an appropriate choice of the free parameters in the scalar potential. These adjustments do not affect the physical observables in the quark and lepton sectors, which are determined mainly by the Yukawa couplings.

The full symmetry group $\mathcal{G}$ exhibits the following spontaneous breaking pattern:

[TABLE]

Here $v=246$ GeV is the electroweak symmetry breaking scale, and we assume that the Pati-Salam gauge symmetry is broken at the scale \Lambda_{PS}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt\sim}\hss}\raise 1.0pt\hbox{\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}}}10^{6} GeV. This restriction follows from the experimental limit on the branching ratio for the rare neutral meson decays, such as $B^{0}\to l_{i}^{\pm}l_{j}^{\mp}$, mediated by the vector leptoquarks, as discussed in Refs. Valencia:1994cj ; Smirnov:2007hv . Furthermore, it is worth mentioning that Pati-Salam models with a quark-lepton unification scale of about \mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt\sim}\hss}\raise 1.0pt\hbox{>}}10^{6} GeV can fulfill gauge coupling unification Hartmann:2014fya . A comprehensive study of gauge coupling unification in models that include all possible chains of Pati-Salam symmetry breaking in both supersymmetric and non-supersymmetric scenarios has been given in Ref. Hartmann:2014fya .

3 Understanding the model setup

In this section we try to motivate in more detail our choice for the model content and the transformation properties. First note that the Pati-Salam gauge symmetry $SU(4)_{C}\otimes SU\left(2\right)_{L}\otimes SU\left(2\right)_{R}$ breaks down to the conventional left-right symmetry $SU(3)_{C}\otimes SU\left(2\right)_{L}\otimes SU\left(2\right)_{R}\otimes U\left(1\right)_{B-L}$ by the VEV of the scalar field $\Sigma$, at the scale \Lambda_{PS}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt\sim}\hss}\raise 1.0pt\hbox{>}}10^{6} GeV. The next symmetry breaking step is triggered by $\chi_{R}$, whose VEV is assumed to be in the few TeV scale, playing an important role in implementing the low-scale seesaw neutrino mass generation Akhmedov:1995ip ; Akhmedov:1995vm . The breaking of the electroweak gauge group $SU(3)_{C}\otimes SU\left(2\right)_{L}\otimes U\left(1\right)_{Y}$ is triggered by the scalar fields $\Phi_{j}$, which acquire vacuum expectation values at the electroweak symmetry breaking scale $v=246$ GeV.

Besides, note that the presence of the scalar field $\Sigma$ transforming as the adjoint representation of $SU\left(4\right)$ is also crucial in the implementation of the Universal Seesaw mechanism for down-type quarks and charged leptons, mediated by exotic fermions. This scalar field $\Sigma$ acquires a VEV at the scale $\Lambda_{PS}$, so that an insertion $\frac{\sigma^{6}}{\Lambda^{6}}$ in its corresponding Yukawa term generates a TeV scale contribution to the exotic charged lepton and exotic down-type quark masses. The charge of $\Sigma$ under the $Z_{16}$ discrete group ensures that its different contributions to the charged leptons and down type quark masses will be comparable to the ones arising from the $A_{i}\overline{\Psi}_{iL}\Psi_{iR}$ ($i=1,2,3$) mass terms (which contribute equally to the down type quark and charged lepton masses). It is worth mentioning that we are assuming $A_{i}\approx\mathcal{O}(1)$ TeV. Let us note that the inclusion of the scalar field $\Sigma$ is necessary to guarantee that the resulting down-type quark masses are different from the charged lepton masses, as it will be shown in Section 4.

Notice that the scalars $\phi_{i}$ and $\varphi_{l}$ are needed to generate the mixing between the standard model charged leptons and down-type quarks with their exotic siblings, so as to implement the Universal Seesaw mechanism that gives rise to realistic masses for the standard model charged fermions. The scalar fields $\chi_{R},\chi_{L}$ and $\xi_{i}$ have Yukawa terms necessary for the implementation of the inverse and linear seesaw mechanisms, so as to generate the light active neutrino masses. This requires also that VEVs of $\xi_{i}$ are much smaller than the electroweak symmetry breaking scale.

The scalar fields $\rho_{i}$ , $\eta_{i}$ and $\tau_{i}$ are needed to generate the diagonal $3\times 3$ blocks that include the mixing of the neutrino states contained in $\overline{F}_{iL}$ and $F_{iR}$ with the singlet neutrinos $S_{i}$ ($i=1,2,3$), thus avoiding the transmission of the strong hierarchy in the up mass matrix to the light active neutrino mass matrix. Furthermore, the scalar field $\sigma$, charged under the $Z_{16}$ discrete group is need to generate the observed SM charged fermion mass and quark mixing hierarchy. In order to relate the quark masses with the quark mixing parameters, we assume that the scalar field $\sigma$ acquires a VEV equal to $\lambda\Lambda$, where $\lambda=0.225$ is one of the Wolfenstein parameters and $\Lambda$ is the cutoff of our model. In summary, the set of VEVs of the scalar fields is assumed to satisfy the following hierarchy:

[TABLE]

where $\langle\chi_{L,R}\rangle=v_{L,R}$. We now comment on the possible VEV patterns for the $\Delta(27)$ scalar triplets $\rho$, $\eta$, $\tau$ and $\xi$. Since the VEVs of the $\Delta(27)$ scalar triplets satisfy the hierarchy $v_{\xi}\ll v_{\rho}=v_{\eta}=v_{\tau}\sim\Lambda_{PS}$, the mixing angles of $\xi$ with $\rho$, $\eta$ and $\tau$ are very tiny since they are suppressed by the ratios of their VEVs, and consequently the method of recursive expansion proposed in Ref. Grimus:2000vj can be used for the analysis of the model scalar potential. In this scenario, the scalar potential for the $\Delta(27)$ scalar triplet $\xi$ can be treated independently from the scalar potential for the $\Delta(27)$ scalar triplets $\rho$, $\eta$, $\tau$, as shown in detail in the appendices B and C. One can see that the following VEV patterns for the $\Delta(27)$ scalar triplets are consistent with the scalar potential minimization equations for a large region of parameter space

[TABLE]

We now turn our attention on the role of each discrete group factor of our model. As will be seen in Sects. 4 and 5 the $\Delta\left(27\right)$ discrete group is crucial for the predictivity of our model, giving rise to viable textures for the fermion masses and mixings. Notice that the $\Delta(27)$ discrete group is a non trivial group of the type $\Delta(3n^{2})$ for $n=3$, isomorphic to the semi-direct product group $(Z_{3}^{\prime}\times Z_{3}^{\prime\prime})\rtimes Z_{3}$ Ishimori:2010au . Recently, this group has been used in multi-Higgs doublet models Bhattacharyya:2012pi , $SO(10)$ models Bjorkeroth:2015uou , warped extra dimensional models Chen:2015jta and models based on the $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}$ gauge symmetry Vien:2016tmh ; Hernandez:2016eod . We introduce the $Z_{16}$ discrete group, since it is the smallest cyclic symmetry that allows the Yukawa term $\overline{F}_{1L}\Phi_{1}F_{1R}\frac{\sigma^{8}}{\Lambda^{8}}$ of dimension twelve from a $\frac{\sigma^{8}}{\Lambda^{8}}$ insertion on the $\overline{F}_{1L}\Phi_{1}F_{1R}$ operator. This leads to the required $\lambda^{8}$ suppression needed to naturally explain the smallness of the up quark mass. The $Z_{16}$ group has been recently shown to be useful for explaining the observed SM charged fermion mass and quark mixing hierarchy, in the framework of a $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}$ models based on the $A_{4}$ and $S_{3}$ family symmetries Hernandez:2015tna ; CarcamoHernandez:2017kra ; Hernandez:2013hea . As we will see in the next section, in our model the CKM matrix arises from the down-type quark sector. In order to get the correct hierarchy in the entries of the quark mass matrices yielding a realistic pattern of quark masses and mixing angles, we use a $Z_{4}$ discrete symmetry and the scalar bidoublets $\Phi_{j}$ ($j=1,2$), one neutral and the another charged under $Z_{4}$. This group was previously used in some other flavor models and proved to be helpful, in particular, in the context of Grand Unification Emmanuel-Costa:2013gia ; Arbelaez:2015toa , models with extended $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}$ gauge symmetry Hernandez:2014vta and warped extradimensional models Hernandez:2015zeh . The $Z_{4}$ is the smallest cyclic symmetry that guarantees that the renormalizable Yukawa terms for the fermion singlets $S_{i}$ ($i=1,2,3$) only involve the scalar fields $\xi_{i}$, assumed to acquire very small VEVs. This feature is crucial to obtain an inverse seesaw contribution to the light active neutrino mass matrix, instead of a double seesaw contribution, thus giving rise to heavy quasi-Dirac neutrinos within the LHC reach.

It is worth noting that the Yukawa Lagrangian (2) possesses accidental $U_{1}$-symmetries located in the non-SM sector with the field charge ($Q$) assignments:

[TABLE]

These are spontaneously broken by the VEVs of the corresponding scalar fields in Eq. (14). As a result there appear massless Goldstone bosons with interaction strengths determined by the VEVs shown in Eq. (14). This leads to the presence of invisible Higgs decays joshipura:1992hp which are restricted by LEP as well as LHC searches Bonilla:2015jdf . From Eq. (13) one sees that the $U_{1}^{(a)}$ symmetry breaking scale is large of the order of $\Lambda_{PS}\sim 10^{6}$ GeV, while the $U_{1}^{(b)}$ symmetry is broken at the low scale $v_{\xi}\sim v_{\zeta}$. Thus the $U_{1}^{(b)}$-Goldstone can have a significant couplings with the fields in the exotic sector and these interactions could potentially leak via mixing to the SM sector. On the other hand, as previously mentioned in section 2, the 125 GeV Higgs boson is dominantly composed of $\left(\Phi_{1}\right)_{11}$, which is the only scalar contributing to the top quark mass. Since the $U_{1}^{(b)}$ symmetry breaking scale $v_{\xi}\sim v_{\zeta}$ is much smaller than the scale of electroweak symmetry breaking scale $v=246$ GeV, the mixing of the $U_{1}^{(b)}$-Goldstone with the 125 GeV Higgs boson is suppressed by the ratios of their VEVs (c.f. Ref. Grimus:2000vj ). Alternatively, these Goldstones may also be avoided by adding explicit breaking trilinear terms in the scalar potential. A detailed study is beyond the scope of the present paper.

4 Quark masses and mixings

From the first line in Eq. (2), the up-type quark mass matrix is given by

[TABLE]

where $y_{i}$ ($i=1,2,3$) are $\mathcal{O}(1)$ parameters and we set $v_{1}^{\left(1\right)}=v_{1}^{\left(2\right)}=\frac{v}{\sqrt{2}}$, with $v=246$ GeV the scale of electroweak symmetry breaking and $v_{\sigma}=\lambda\Lambda$, with $\lambda=0.225$ being one of the Wolfenstein parameters.

For the sake of simplicity, we assume $v_{2}^{\left(j\right)}=0$ ($j=1,2$) so that the standard model down-type quarks and charged leptons acquire their masses from a universal seesaw mechanism, mediated by the exotic down-type quarks $D_{i}$ and charged leptons $L_{i}$ present in $\Psi_{iR}$ and $\overline{\Psi}_{iL}$. In this case the down-type charged fermion mass matrices take the form

[TABLE]

with

[TABLE]

and the further simplification $v_{\phi_{i}}=v_{\phi}$ and $v_{\varphi_{l}}=v_{\varphi}$.

Taking the limit $M_{a}$, $M_{b}\ll A_{i}$, the standard model down-type quark and charged lepton mass matrices become

[TABLE]

where we have set $v_{\phi}=\lambda^{3}\frac{v}{\sqrt{2}}\frac{m_{\Psi}}{v_{\varphi}}$.

Notice that in our model the CKM matrix arises only from the down-type quark sector. In order to recover the low energy quark flavor data, we assume that all dimensionless parameters of the SM down-type quark mass matrix are real, excepting $a_{5D}$, taken to be complex. The physical quark mass spectrum Bora:2012tx ; Xing:2007fb and mixing angles Olive:2016xmw can be perfectly reproduced in terms of natural parameters of order one, as shown in Table 4, starting from the following benchmark point:

[TABLE]

To close this section we briefly comment on the phenomenological implications of our model regarding flavor changing processes involving quarks. It is noteworthy that the flavor changing top quark decays $t\to hc$ and $t\to hu$ are absent at tree level in our model, as follows from the quark Yukawa terms. The flavor changing top quark decays $t\to hc$ and $t\to hu$ are induced at one loop level from virtual charged Higgses and SM down-type quarks running in the internal lines of the loops. Consequently, a measurement of the branching fraction for the $t\to hc$ and $t\to hu$ decays at the LHC may test our model. On the other hand the admixture of exotic down-type quarks in the mass matrix implies a violation of the Glashow-Iliopoulos-Maiani mechanism in this sector. This may be relevant in connection with the recent B anomalies Bifani:2017gyn .

5 Lepton masses, mixing and oscillations

Here is where the predictive power of our flavor symmetry model is mainly manifest. From the neutrino Yukawa terms, we obtain the following neutrino mass terms :

[TABLE]

where the neutrino mass matrix reads

[TABLE]

with

[TABLE]

where for the sake of simplicity, we set $\alpha_{i}=\beta_{i}=y_{i}$ ($i=1,2,3$) and $v_{\rho}=v_{\eta}=v_{\tau}=\frac{v}{\sqrt{2}v_{R}}\Lambda$. Then,

[TABLE]

where $M_{\nu}^{\left(1\right)}$ corresponds to the physical light neutrino mass matrix whereas $M_{\nu}^{\left(2\right)}$ and $M_{\nu}^{\left(3\right)}$ are the heavy quasi-Dirac neutrino mass entries. Note that the physical eigenstates include three active neutrinos and six heavy, mainly isosinglet, neutrinos. The heavy quasi-Dirac neutrinos have a small splitting $\mu$.

In the limit of vanishing contributions from linear seesaw ($v_{L}\ll v_{R}$), the light neutrino mass matrix becomes

[TABLE]

Taking real Yukawa couplings $\lambda_{i}$ and VEVs, $M_{\nu}^{\left(1\right)}$ display explicit generalized $\mu-\tau$ symmetry babu:2002dz ; grimus:2003yn ; King:2014nza

[TABLE]

with

[TABLE]

The most general matrix $V_{\nu}$ that diagonalizes $M_{\nu}^{\left(1\right)}$ through $V_{\nu}^{T}M_{\nu}^{\left(1\right)}V_{\nu}=\mathrm{diag}(m^{\nu}_{1},m^{\nu}_{2},m^{\nu}_{3})$ is given by Chen:2015siy ; Chen:2016ica

[TABLE]

where

[TABLE]

is the Takagi factorization matrix of $X$ defined as $X=\Sigma\Sigma^{T}$ , $O_{ij}$ are $3\times 3$ orthogonal matrix parameterized as

[TABLE]

and $Q_{\nu}=\mathrm{diag}(e^{-i\pi k_{1}/2},e^{-i\pi k_{2}/2},e^{-i\pi k_{3}/2})$ is a diagonal matrix with $k_{i}=0,1,2,3$.

Notice that, with real Yukawa couplings $\lambda_{i}$ and VEVs in Eq.(49), we have a reduced number of parameters, leading to the following relations:

[TABLE]

On the other hand, the light charged lepton mass matrix in Eq. (34) is diagonalized by an almost diagonal unitary matrix through $V_{l}^{\dagger}M^{\left(1\right)}_{l}M_{l}^{\left(1\right)\dagger}V_{l}=\mathrm{diag}(m^{2}_{e},m^{2}_{\mu},m^{2}_{\tau})$. Taking real entries in $M^{\left(1\right)}_{l}$, at first approximation $V_{l}$ is dominated by the Cabibbo angle and can be written as

[TABLE]

In this approximation, the lepton mixing matrix is given as

[TABLE]

In the fully “symmetrical” presentation of the lepton mixing matrix Schechter:1980gr ; Rodejohann:2011vc

[TABLE]

with $c_{ij}=\cos\theta_{ij}$ and $s_{ij}=\sin\theta_{ij}$, the relation between flavor mixing angles and the magnitudes of the entries of the leptonic mixing matrix is

[TABLE]

The Jarlskog invariant is defined as $J_{\mathrm{CP}}=\mathrm{Im}\left(U_{11}^{*}U_{23}^{*}U_{13}U_{21}\right)$, and takes the form Rodejohann:2011vc

[TABLE]

whereas the two additional Majorana-type rephasing invariants $I_{1}=\mathrm{Im}\left(U_{12}^{2}U_{11}^{*2}\right)$ and $I_{2}=\mathrm{Im}\left(U_{13}^{2}U_{11}^{*2}\right)$, become

[TABLE]

In terms of the model parameters, the lepton mixing angles are expressed as

[TABLE]

[TABLE]

[TABLE]

and our predicted correlations:

[TABLE]

[TABLE]

On the other hand, the rephasing invariant CP violation parameter combinations are

[TABLE]

[TABLE]

[TABLE]

Eliminating $\omega_{12}$ in the above relations using eq.(56), the mixing parameters depend ultimately on three angles $\omega_{23}$, $\omega_{13}$, $\psi$ up to three discrete variables $k_{1}$, $k_{2}$, $k_{3}$. Furthermore, without loss of generality these angles are restricted to $\omega_{23}\in[-\pi,\pi]$, $\omega_{13}\in[-\pi/2,\pi/2]$ and $\psi\in[-\pi/2,\pi/2]$. Notice that the angle $\psi$ in this framework is responsible for the CP violating phase in the lepton sector, since the first stage of the diagonalization process yields a real symmetric matrix

[TABLE]

and in the limit of vanishing $\psi$, this matrix is already block diagonal, implying $\omega_{23}=\omega_{13}=0$ in Eq.(68), which in turn leads to $J_{\mathrm{CP}}=0$. Moreover, from Eq.(66), notice that if $\sin^{2}\theta_{13}$ is allowed to vary within $3\sigma$ according to the global fit Forero:2014bxa , then $\sin^{2}\theta_{23}$ is restricted to the range $(0.487,0.539)$.

In figures 1 and 2, we give the allowed values for $\sin^{2}\theta_{23}$ together with the corresponding $J_{\text{CP}}$ predictions in both mass orderings. For our analysis, we randomly generated parameter configurations for $\omega_{23}$, $\omega_{13}$ and $\psi$ corresponding to 3(1)$\sigma$ values for the solar and reactor angles in Eqs. (63),(64). One sees that for the Normal Hierarchy (NH) case the allowed region is severely restricted, with a fourfold degeneracy. In this case CP must necessarily be violated in oscillations, and the predicted atmospheric angle lies in the higher octant, inside its $1\sigma$ region.

In contrast, for the case of Inverted Hierarchy (IH) one sees that CP can be conserved in neutrino oscillations. Moreover, if violated, it is unlikely for CP to be maximally violated. The predicted atmospheric angle lies inside its $2\sigma$ region, preferably in the first octant.

6 Neutrinoless double beta decay

In this section we determine the effective Majorana neutrino mass parameter characterizing the neutrinoless double beta ($0\nu\beta\beta$) decay amplitude. It is given by:

[TABLE]

where $U_{ei}^{2}$ and $m_{\nu_{i}}$ are the lepton mixing matrix elements and the light active neutrino masses, respectively. The light active neutrino masses can be written in terms of the parameters of the model as

[TABLE]

with

[TABLE]

We show in Figure 3 the effective Majorana neutrino mass parameter $|m_{\beta\beta}|$ versus the lightest active neutrino mass for the cases of normal and inverted neutrino mass hierarchies. In order to determine the predicted ranges for $|m_{\beta\beta}|$ in our model, we have randomly generated the angles $\omega_{23}$, $\omega_{13}$ and $\psi$, as well as the light active neutrino mass scale $m_{\nu}=\frac{\lambda_{2}v_{\xi}}{\sqrt{2+r^{2}}}$ in a range of values where the neutrino mass squared splittings and the leptonic mixing parameters are consistent with the observed neutrino oscillation data. Our predicted range of values for the effective Majorana neutrino mass parameter has a lower bound, even in the case of normal hierarchy, indicating that a complete destructive interference among the three light neutrinos is prevented by our symmetry and the current oscillation data.

The corresponding $\rm 0\nu\beta\beta$ decay rates are within the reach of the next-generation bolometric CUORE experiment Alessandria:2011rc or, more realistically, of the next-to-next-generation ton-scale $0\nu\beta\beta$-decay experiments. It is worth mentioning that the Majorana neutrino mass parameter has an upper bound on $|m_{\beta\beta}|\leq(61-165)$ meV at 90% C.L, as indicated by the KamLAND-Zen experiment from the limit on the ${}^{136}\mathrm{Xe}$ $0\nu\beta\beta$ decay half-life $T^{0\nu\beta\beta}_{1/2}({}^{136}\mathrm{Xe})\geq 1.07\times 10^{26}$ yr KamLAND-Zen:2016pfg . This bound will be improved within a not too far future. The GERDA “phase-II”experiment Abt:2004yk ; Ackermann:2012xja is expected to reach $T^{0\nu\beta\beta}_{1/2}(^{76}{\rm Ge})\geq 2\times 10^{26}$ yr, corresponding to $|m_{\beta\beta}|\leq 100$ meV. A bolometric CUORE experiment, using ${}^{130}\mathrm{Te}$ Alessandria:2011rc , is currently under construction and its estimated sensitivity is close to about $T_{1/2}^{0\nu\beta\beta}(^{130}\mathrm{Te})\sim 10^{26}$ yr, implying $|m_{\beta\beta}|\leq 50$ meV. Furthermore, there are plans for ton-scale next-to-next generation $0\nu\beta\beta$ experiments with 136Xe KamLANDZen:2012aa ; Albert:2014fya and 76Ge Abt:2004yk ; Guiseppe:2011me , asserting sensitivities over $T_{1/2}^{0\nu\beta\beta}\sim 10^{27}$ yr, corresponding to $|m_{\beta\beta}|\sim 12-30$ meV.

7 Discussions and Conclusions

We have proposed a realistic extension of the standard model within the Pati-Salam framework. The theory incorporates a flavor symmetry based on the $\Delta\left(27\right)$ group and realizes a realistic Froggatt-Nielsen picture of quark mixing. Concerning the lepton sector, neutrino masses arise from an inverse seesaw mechanism and the allowed ranges for the atmospheric mixing angle $\theta_{23}$ and CP violating phase $\delta_{CP}$ are rather restricted once one takes into account the precise measurements of the remaining oscillation parameters. This makes the model rather predictive. Our main neutrino oscillation results are summarized in Figs. 1 and 2. We find that, for normal neutrino mass ordering, the atmospheric angle must lie in the higher octant and that CP must be violated in oscillations. In contrast, for inverse hierarchy, the lower octant is favored and the range of allowed Jarlskog invariant extends from zero up to a non-maximal value. Our results concerning $\rm 0\nu\beta\beta$ decay are summarized in Fig.3. They indicate the existence of a lower bound for the $\rm 0\nu\beta\beta$ decay rate, a feature also encountered in other flavor models Dorame:2011eb ; Dorame:2012zv ; King:2013hj ; Bonilla:2014xla ; Gehrlein:2016wlc . As mentioned, neutrino masses arise from a low-scale seesaw mechanism, whose messengers may be produced at the LHC either through a charged or neutral gauge portal Das:2012ii ; Deppisch:2013cya ; Queiroz:2016gif . Admittedly, the model is rather complex, especially in its scalar sector. However, it serves as a “proof-of-concept” attempt, namely, to our knowledge this is the first time that a fully realistic and to some extent predictive flavor realization of a Pati-Salam scenario is given.

Acknowledgments

This research is supported by the Spanish grants FPA2014-58183-P, Multidark CSD2009-00064, SEV-2014-0398 (MINECO) and PROMETEOII/2014/084 (Generalitat Valenciana); Chilean grants Fondecyt No. 1170803, No. 1150792 and CONICYT PIA/Basal FB0821 and ACT1406; and Mexican grant CONACYT No. 274397. A.E.C.H is very grateful to Professor Hoang Ngoc Long and the other members of its group for the warm hospitality of the Institute of Physics, Vietnam Academy of Science and Technology, where this work was completed.

Appendix

Appendix A Product rules of the $\Delta(27)$ discrete group

The $\Delta(27)$ discrete group is a subgroup of $SU(3)$, has 27 elements divided into 11 conjugacy classes, so that it has 11 irreducible representations. These irreducible representations are: two triplets, i.e., $\mathbf{3}_{[0][1]}$ (which we denote by $\mathbf{3}$) and its conjugate $\mathbf{3}_{[0][2]}$ (which we denote by $\overline{\mathbf{3}}$) and 9 singlets, i.e., $\mathbf{1}_{k,l}$ ($k,l=0,1,2$), where $k$ and $l$ correspond to the $Z_{3}$ and $Z_{3}^{\prime}$ charges, respectively Ishimori:2010au . The $\Delta(27)$ discrete group is a simple group of the type $\Delta(3n^{2})$ with $n=3$ and is isomorphic to the semi-direct product group $(Z_{3}^{\prime}\times Z_{3}^{\prime\prime})\rtimes Z_{3}$ Ishimori:2010au . Indeed, the simplest group of the type $\Delta(3n^{2})$ is $\Delta(3)\equiv Z_{3}$. The next group is $\Delta(12)$, which is isomorphic to $A_{4}$. Thus, the $\Delta(27)$ discrete group is the next simplest nontrivial group of the type $\Delta(3n^{2})$. It is worth mentioning that one can write any element of the $\Delta(27)$ discrete group as $b^{k}a^{m}{a^{\prime}}^{n}$, where $b$, $a$ and $a^{\prime}$ correspond to the generators of the $Z_{3}$, $Z_{3}^{\prime}$ and $Z_{3}^{\prime\prime}$ cyclic groups, respectively. These generators fulfill the relations:

[TABLE]

The characters of the $\Delta(27)$ discrete group are shown in Table 5. Here $n$ is the number of elements, $h$ is the order of each element, and $\omega=e^{\frac{2\pi i}{3}}=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$ is the cube root of unity, which satisfies the relations $1+\omega+\omega^{2}=0$ and $\omega^{3}=1$. The conjugacy classes of $\Delta(27)$ are given by:

[TABLE]

The tensor products between $\Delta(27)$ triplets are described by the following relations Ishimori:2010au :

[TABLE]

where we introduced the shorthand notation $\mathbf{3}_{[0][1]}\equiv\mathbf{3}$ and $\mathbf{3}_{[0][2]}\equiv\mathbf{\bar{3}}$ used in Eq.(2). In the above formulas $\mathbf{3}_{A}$ and $\mathbf{3}_{S_{1,2}}$ are an antisymmetric and two variants of symmetric triplets. The multiplication rules between $\Delta(27)$ singlets and $\Delta(27)$ triplets are given by Ishimori:2010au :

[TABLE]

The tensor products of $\Delta(27)$ singlets $\mathbf{1}_{k,\ell}$ and $\mathbf{1}_{k^{\prime},\ell^{\prime}}$ take the form Ishimori:2010au :

[TABLE]

From the equation given above, we obtain explicitly the singlet multiplication rules of the $\Delta(27)$ group, which are given in Table 6.

Appendix B Scalar potential for one $\Delta(27)$ scalar triplet

The scalar potential for a $\Delta(27)$ scalar triplet, i.e., $\xi$ is given by:

[TABLE]

Assuming the VEV configuration for the $\Delta(27)$ scalar triplet $\xi$:

[TABLE]

We find that the scalar potential minimization equations are:

[TABLE]

Then, from the scalar potential minimization equations, we find the following relations:

[TABLE]

Thus, the VEV pattern for the $\Delta(27)$ triplet, i.e., $\xi$ given by Eq. (82), is compatible with a global minimum of the scalar potential of Eq. (81) for a large region of parameter space.

Appendix C Scalar potential for three $\Delta(27)$ scalar triplets

The scalar potential for the three $\Delta(27)$ scalar triplets $\rho$, $\tau$ and $\eta$ is

[TABLE]

where $V_{\rho}$, $V_{\tau}$ and $V_{\eta}$ are the scalar potentials for the $\Delta(27)$ scalar triplets $\rho$, $\tau$ and $\eta$, respectively, whereas $V_{\rho,\tau}$, $V_{\tau,\eta}$ and $V_{\rho,\eta}$ describe interaction terms involving the pairs ($\rho$, $\tau$), ($\tau$, $\eta$) and ($\rho$, $\eta$). The different parts of the scalar potential for the three $\Delta(27)$ scalar triplets take the form:

[TABLE]

Now we determine the conditions under which the VEV pattern for the $\Delta\left(27\right)$ scalar triplets is a solution of the scalar potential given in Eq. (85). In view of the very large number of parameters of the scalar potential for the $\Delta\left(27\right)$ scalar triplets and in order to simplify the analysis, we assume universality in its trilinear and quartic couplings, i.e.

[TABLE]

Considering the VEV alignment

[TABLE]

We find the following scalar potential minimization equations:

[TABLE]

Then, from the scalar potential minimization equations, we find the following relations:

[TABLE]

Consequently, the VEV patterns for the three $\Delta(27)$ triplet scalars given by Eq. (93) are compatible with a global minimum of the scalar potential of Eq. (85) for a large region of parameter space.