= Cubic superpotential with right-handed-neutrino superfields
For the <MSSM superfield representations> augmented by singlet <right-handed neutrino> <chiral superfields> $N_i$, the homogeneous cubic <gauge-invariant> <superpotential> contains the four Yukawa structures $QH_uU$, $QH_dD$, $LH_dE$, $LH_uN$ and the additional structures $LLE$, $LQD$, $UDD$, $NH_dH_u$, $NNN$. Weak-doublet pairs contract with the antisymmetric two-index tensor; $UDD$ contracts with the color three-index tensor. Consequently the $LLE$ coefficient is antisymmetric in its two $L$ indices, the $UDD$ coefficient in its two $D$ indices, and the $NNN$ coefficient symmetric. With $L(N)=-1$, the last five structures violate <baryon number> or <lepton number>, whereas the first four conserve both. Singlet fields must not be omitted when classifying a general cubic <superpotential>.
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