in Eq. (\[leprt\]) to form a vector field $l_{L+R,3}$ so as to insure gauge anomaly cancellation. Similarly, the Higgs scalars decompose as $$S_{4} \rightarrow S_{1}(\textbf{1}, +1, +1) + S_{3}(\textbf{3}, +1, +1) \;\;,$$ $$H_{4} \rightarrow H_{1} (\textbf{1}, +1, \omega^2) + H_{3} (\textbf{3}, +1, \omega^2) \;\;,$$ $$H'_{4} \rightarrow H'_{1} (\textbf{1}, -1, \omega^2) + H'_{3} (\textbf{3}, -1, \omega^2) \;\;,$$ and $$\Phi_{3} \rightarrow \Phi_{3}(\textbf{3}, +1, \omega^2) \;\;.$$ After this stage of $A_{5} \rightarrow A_{4}$ breaking via $\langle S_{4} \rangle$, the Yukawa Lagrangian can be written as $$\begin{aligned}
\label{A4potential}
L^{A_{4}}_{Yukawa} &=& \frac{1}{2}M_{1}N^{(1)}_{R}N^{(1)}_{R} + \frac{1}{2}M_{5}(N^{(2)}_{R}N^{(3)}_{R} + N_{R3}N_{R3}) \nonumber \\
&+& Y_{S1} \left[(V_{S} + S_{1})(N^{(2)}_{R}N^{(3)}_{R} + N_{R3}N_{R3}) + S_{3}(N_{R3}N_{R3}) \right] \nonumber \\
&+& Y_{S2}\left[(V_{S} + S_{1})(l_{R3})^{c}l_{R3} + S_{3}(l_{R3})^{c}(l_{R3} + l_{R1'} + l_{R1''}) \right] \nonumber \\
&+& Y_{1}\left[L_{L1}N^{(1)}_{R}H_{1} + L_{L3}N_{R}^{(1)}H_{3} \right] \nonumber \\
&+& Y_{2}\left[L_{L1}N_{R3}H_{3} + L_{L3}N_{R3}(H_{1} + H_{3}) + L_{L3}(N^{(2)}_{R} + N^{(3)}_{R})H_{3}\right] \nonumber \\
&+& Y_{3}\left[L_{L1}N_{R3}\Phi_{3} + L_{L3}N_{R3}\Phi_{3} + L_{L3}(N^{(2)}_{R} + N^{(3)}_{R})\Phi_{3}\right] \nonumber \\
&+& Y_{4}\left[L_{L1}l_{R3}H'_{3} + L_{L3}l_{R3}(H'_{1} + H'_{3}) + L_{L3}(l_{R1'} + l_{R1''})H'_{3}\right] \nonumber \\
&+& Y_{5}\left[L_{L1}l^{(1)}_{R1}H'_{1} + L_{L3}l^{(1)}_{R1}H'_{3} \right] + Y_{6}\left[L_{L1}l^{(2)}_{R1}H'_{1} + L_{L3}l^{(2)}_{R1}H'_{3}\right] + {\rm H.c.}\end{aligned}$$ For convenience, the multiplication rules for $A_4$ are summarized in Appendix B.

Phenomenology
=============

We begin with the mass spectrum of the leptons.

Charged Lepton Masses
---------------------

After the 