FCNCs, proton stability, $g_{\mu } - 2$ discrepancy, neutralino cold dark matter in flipped $SU(5) \times U(1)_{\chi }$ from F theory with $A_{4}$ symmetry

Abstract

We predict the low energy signatures of a Flipped $SU(5) \times U(1)_{\chi }$ effective local model, constructed within the framework of F-theory based on $A_{4}$ symmetry. The Flipped SU(5) model from F Theory in the field of particle physics is prominent due to its ability to construct realistic four-dimensional theories from higher-dimensional compactifications which necessitates a unified description of the fundamental forces and particles of nature, used for exploring various extensions of the Standard Model. We study Flipped $SU(5) \times U(1)_{\chi }$ Grand Unified Theories (GUTs) with $A_{4}$ modular symmetry. In our model due to different modular weights assignments, the fermion mass hierarchy exists with different weighton fields. The constraints on the Dirac neutrino Yukawa matrix allows a good tuning to quark and charged lepton masses and mixings for each weighton field, with the neutrino masses and lepton mixing well determined by the type I seesaw mechanism which occurs at the expense of some tuning which manifests itself in charged lepton flavor violating decays which we explore here. The minimal Flipped SU(5) model is supplemented with an extra right-handed type and its complex conjugate electron state, $E_{c} + \bar{E_{c}}$, as well as neutral singlet fields. The $E_{c} + \bar{E_{c}}$ pair gets masses of the order of TeV which solves the $g_{\mu }- 2$ discrepancy. The predictions of the model for charged lepton flavor violation decay rate and proton decay could be tested in near future experiments. Also we detect in our model the existence of neutralino, its charge mass and spin via direct and indirect detection.

Appendix A: \(SU(5)\times U(1)\) Symmetry

In our pursuit of understanding the flipped $SU(5)\times U(1)$ model within a generic F-theory framework, we adopt the spectral cover approach and introduce fluxes along U(1) factors to elucidate the geometric properties of matter curves and the massless particle spectrum associated with them. Through this comprehensive analysis, we successfully identify the presence of three generations of chiral matter fields and ascertain the requisite Higgs representations necessary for breaking the symmetry, thereby establishing a robust foundation for further investigations into the model’s low-energy implications and phenomenological predictions. Within each family, the chiral matter fields form a comprehensive 16 spinorial representation of SO(10), amenable to the insightful $SU(5)\times U(1)_{\chi }$ decomposition [36, 37].

$$\begin{aligned} 16 = 10_{-1} + \bar{5_{3}} + 1_{-5} \end{aligned}$$

(A1)

Consequently, the Standard Model representations find their embedding as follows:

$$\begin{aligned}{} & {} 10_{-1} \Longrightarrow F_{i} = \left( Q_{i}, d_{i}^{c}, \nu _{i}^{c} \right) \end{aligned}$$

(A2)

$$\begin{aligned}{} & {} {\bar{5}}_{+3} \Longrightarrow {\bar{f}}_{i} = \left( u_{i}^{c}, l_{i} \right) \end{aligned}$$

(A3)

$$\begin{aligned}{} & {} 1_{-5} \Longrightarrow l_{i}^{c} = e_{i}^{c} \end{aligned}$$

(A4)

The spontaneous breaking of the flipped SU(5) symmetry unfolds through the utilization of a pair of accommodated Higgs fields.

$$\begin{aligned}{} & {} H=10_{-1}=\left( Q_{H},d_{H}^{c},\nu _{H}^{c}\right) \end{aligned}$$

(A5)

$$\begin{aligned}{} & {} {\bar{H}} = {\bar{10}}_{+1}= \left( {\bar{Q}}_{H},{\bar{d}}_{H}^{c},{{\bar{\nu }}}_{H}^{c}\right) \end{aligned}$$

(A6)

The representation of the MSSM Higgs doublets as fiveplets can be traced back to their origin within the SO(10) group’s 10-dimensional representation.

$$\begin{aligned} h = 5_{+2} = \left( D_{h},h_{d}\right) , {\bar{h}} = {\bar{5}}_{-2}= \left( D_{h},h_{u}\right) \end{aligned}$$

(A7)

This $U(1)_{\chi }$ charge assignment not only distinguishes between the Higgs $\bar{5_{-2}}$ fields and the matter anti-fiveplets $\bar{5_{-3}}$ in the flipped model, but it also plays a crucial role in generating fermion masses through $SU(5)\times U(1)_{\chi }$ invariant couplings. The generation of fermion masses is attributed to the interaction terms involving $SU(5) \times U(1)_{\chi }$ invariant couplings.

$$\begin{aligned} W= & {} \lambda _{d}10_{-1}.10_{-1}.5^{h}_{2} + \lambda _{u}10_{-1}.{\bar{5}}_{3}.{\bar{5}}^{{\bar{h}}}_{-2} + \lambda _{d}Q.d^{c}.{h}_{d} \nonumber \\{} & {} + \lambda _{u}\left( Q u^{c}h_{u} + l\nu ^{c}h_{u}\right) + \lambda _{l}e^{c}lh_{d} \end{aligned}$$

(A8)

Notably, the GUT-scale predictions of the flipped model establish a distinct relationship between up-quark and neutrino Dirac mass matrices, characterized by $m_{t} = m_{\nu _{D}}$. However, contrary to the standard SU(5) model, the flipped model introduces a disparity in the origins of down quark and lepton mass matrices due to their dependence on separate Yukawa couplings. Shifting focus to the Higgs sector, the acquisition of significant vacuum expectation values (VEVs) by H and ${\bar{H}}$ of the order MGUT induces the breaking of $SU(5) \times U(1)_{\chi }$ to the Standard Model gauge group, while simultaneously conferring substantial masses upon the color triplets, as evidenced by the ensuing mass terms.

$$\begin{aligned} HHh + {\bar{H}}{\bar{H}}{\bar{h}} \Rightarrow<\nu _{H}^{c}>d_{H}^{c}D + <\bar{\nu _{H}^{c}}>\bar{d_{H}^{c}}{\bar{D}} \end{aligned}$$

(A9)

Furthermore, an additional higher-order term responsible for imparting Majorana masses to right-handed neutrinos takes the following form:

$$\begin{aligned} \textit{W} = \frac{1}{M_{s}}\bar{10_{{\bar{H}}}}\bar{10_{{\bar{H}}}}10_{-1}10_{-1} \Rightarrow \frac{1}{M_{s}} <\bar{\nu ^{c}_{H}}^{2}\nu _{i}^{c}\nu _{j}^{c} \end{aligned}$$

(A10)