Flipped SU(5) - Superpotential

Superpotential

A generic invariant renormalizable superpotential is a (complex) invariant cubic polynomial in the superfields which has an R-charge of 2. It is a linear combination of the following terms: \begin{matrix} S&S\\ S 10_H \overline{10}_H&S 10_H^{\alpha\beta} \overline{10}_{H\alpha\beta}\\ 10_H 10_H H_d&\epsilon_{\alpha\beta\gamma\delta\epsilon}10_H^{\alpha\beta}10_H^{\gamma\delta} H_d^{\epsilon}\\ \overline{10}_H\overline{10}_H H_u&\epsilon^{\alpha\beta\gamma\delta\epsilon}\overline{10}_{H\alpha\beta}\overline{10}_{H\gamma\delta}H_{u\epsilon}\\ H_d 10 10&\epsilon_{\alpha\beta\gamma\delta\epsilon}H_d^{\alpha}10_i^{\beta\gamma}10_j^{\delta\epsilon}\\ H_d \bar{5} 1 &H_d^\alpha \bar{5}_{i\alpha} 1_j\\ H_u 10 \bar{5}&H_{u\alpha} 10_i^{\alpha\beta} \bar{5}_{j\beta}\\ \overline{10}_H 10 \phi&\overline{10}_{H\alpha\beta} 10_i^{\alpha\beta} \phi_j\\ \end{matrix}

The second column expands each term in index notation (neglecting the proper normalization coefficient). i and j are the generation indices. The coupling Hd 10i 10j has coefficients which are symmetric in i and j.

In those models without the optional φ sterile neutrinos, we add the nonrenormalizable couplings instead.

\begin{matrix} (\overline{10}_H 10)(\overline{10}_H 10)&\overline{10}_{H\alpha\beta}10^{\alpha\beta}_i \overline{10}_{H\gamma\delta} 10^{\gamma\delta}_j\\ \overline{10}_H 10 \overline{10}_H 10&\overline{10}_{H\alpha\beta}10^{\beta\gamma}_i\overline{10}_{H\gamma\delta}10^{\delta\alpha}_j \end{matrix}

These couplings do break the R-symmetry.

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