where H$_{u}$, H$_{d}$, Q, L, u, d, e in Eq. (103) are chiral superfields, and the dimensionless Yukawa couplings λ$_{u}$, λ$_{d}$, λ$_{e}$ are 3 × 3 matrices in the family space. We have suppressed the gauge and family indices. The H$_{u}$, H$_{d}$, Q, L fields are SU (2) doublets, while u, d, e are SU (2) singlets. The last term is the µ term, which is a SUSY version of the SM Higgs boson mass. Terms proportional to H ∗ $_{u}$H$_{u}$ or H ∗ $_{d}$H$_{d}$ are forbidden in the superpotential, since W$_{MSSM}$ must be analytic in the chiral fields. H$_{u}$ and H$_{d}$ are required not only because they give masses to all the quarks and leptons, but also for the cancellation of gauge anomalies. The Yukawa matrices determine the masses and CKM mixing angles of the ordinary quarks and leptons through the neutral components of H$_{u}$ = ( H + u , H 0 $_{u}$) and H$_{d}$ = ( H 0 $_{$_{d}$H}$− d ). Since the top quark, bottom quark and tau lepton are the heaviest fermions in the SM, we assume that only the third family, (3 , 3) element of the matrices λ$_{u}$, λ$_{d}$, λ$_{e}$ are important. and contributes to the Higgs ( mass ) 2 terms in the scalar potential through Note that Eq. (105) is positive definite. Therefore, it cannot lead to electroweak symmetry breaking without including SUSY breaking ( mass ) 2 soft terms for the Higgs fields, which can be negative. Hence, | µ | 2 should almost cancel the negative soft ( mass ) 2 term in order to allow for a Higgs VEV of order ∼ 174 GeV. That the two different sources of masses should be precisely of same order is a puzzle for which many solutions has been suggested [362–365]. The most general gauge invariant and renormalizable superpotential would also include baryon number B or lepton number L violating terms, with each violating by one unit: W$_{Δ}$$_{L}$$_{=1}$ = 1 $_{2}$λ $^{ijk}$L$_{i}$ L$_{j}$e$_{k}$ + λ ′ $^{ijk}$L$_{i}$ Q$_{j}$d$_{k}$ + µ ′ $^{i}$L$_{i}$ H$_{µ}$ and W$_{Δ}$$_{B}$$_{=1}$ = 1 $_{2}$λ ′′ $^{ijk}$u$_{i}$ d$_{j}$d$_{k}$ , where i = 1 , 2 , 3 represents the family indices. The chiral supermultiplets carry baryon number assignments B = +1 / 3 for Q$_{i}$ , B = − 1 / 3 for u$_{i}$, d$_{i}$ , and B = 0 for all others. The total lepton number assignments are L = +1 for L$_{i}$ , L = − 1 for e$_{i}$ , and L = 0 for all the others. Unless λ ′ and λ ′′ terms are very much suppressed, one would obtain rapid proton decay which violates both B and L by one unit. There exists a discrete Z$_{2}$ symmetry, which can forbid baryon and lepton number violating terms, known as R -parity [366]. For each particle: The µ term provides masses to the Higgsinos L ⊃ − µ ( ˜ H + u ˜ $_{H}$− d − ˜ H 0 u ˜ H 0 $_{d}$) + c.c , (104) − L ⊃ V ⊃ | µ | $^{2}$( | H 0 u | 2 + | H + u | 2 + | H 0 d | 2 + | $_{H}$− d | $^{2}$) . (105) P$_{R}$ = ( − 1) 3( B − L )+2 s (106) 48