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In the MSSM, each fermion generation has chiral superfields $Q:(3,2,1/6)$, $U^c:(\bar{3},1,-2/3)$, $D^c:(\bar{3},1,1/3)$, $L:(1,2,-1/2)$, and $E^c:(1,1,1)$. This uses hypercharge normalized by $Q_{\mathrm{electric}}=T^3+Y$ and only left-handed Weyl spinors. Two Higgs chiral doublets have charges $\pm 1/2$. The three vector superfields transform as $(8,1,0)$, $(1,3,0)$ and $(1,1,0)$, supplying the gauge bosons and gauginos.

Use the hypercharge normalization $Q_{\mathrm{electric}}=T^3+Y$. All matter chiral superfields are written with left-handed Weyl spinors, so the fields denoted by a superscript $c$ contain the charge conjugates of the usual right-handed Standard Model fermions. The MSSM superfield representations areEach fermion generation contributes the first five chiral superfields. Their complex scalar field partners are squarks for the quarks and sleptons for the leptons. Each Higgs chiral doublet contains a Higgs complex scalar field and a higgsino. Every chiral superfield also has a complex auxiliary field. The vector superfields areThey contain the corresponding gauge bosons, gauginos in the Adjoint representation, and real auxiliary fields. A right-handed neutrino chiral superfield is not part of the minimal field content.

A gauge anomaly is a quantum obstruction to a classical gauge symmetry. For hypercharge, triangle diagrams with left-handed Weyl spinors can violate the Ward identity of the gauge boson; an uncancelled gauge anomaly makes the gauge theory inconsistent. Anomaly cancellation sums over every component, including colour and weak multiplicities. Complex scalar fields do not contribute to these chiral gauge anomalies. For one fermion generation, the cubic hypercharge coefficient isThe other coefficients involving a hypercharge gauge boson vanish too. With the fundamental index $T(\mathbf{N})=1/2$,The last line is the mixed gauge-gravitational anomaly. Coefficients with one non-Abelian generator and two hypercharge generators vanish by tracelessness. For completeness, the purely colour cubic gauge anomaly cancels between the two fundamental quark components and the two antifundamentals; the weak group has no perturbative cubic gauge anomaly. Its four left-handed doublets per fermion generation also avoid the Witten SU(2) anomaly. Thus each family is separately anomaly-free, not merely their sum.

The gauginos do not spoil this result: their hypercharge is zero and their Adjoint representation is real. One extra Higgs chiral doublet is different because its higgsino is chiral. For $H_u$, its contributions areThey have no compensating contribution from the Higgs complex scalar field. The higgsino in $H_d$ supplies precisely the negative of each coefficient. Opposite-hypercharge Higgs chiral doublets restore anomaly cancellation. The same pair restores an even number of weak fermion doublets, so the Witten SU(2) anomaly provides an additional check of the higgsino anomaly cancellation.

The independent reason is the holomorphic need for two Higgs chiral doublets. A superpotential is a holomorphic function of chiral superfields, so it cannot use a conjugate Higgs superfield to generate the missing Yukawa couplings. The ordinary Standard Model can use a Higgs scalar and its conjugate, but the MSSM needs distinct chiral superfields of both hypercharges. For example,where the dot contracts weak indices with the antisymmetric tensor. All three terms are gauge-invariant holomorphic functions. $H_u$ supplies up-type masses, while $H_d$ supplies down-type and charged-lepton masses. Replacing either by the conjugate of the other would violate the holomorphic closure of chiral superfields.

Use the weak hypercharge normalization $Q=T_3+Y$. Write $U(x)\in SU(2{)}_L$ and the hypercharge transformation as $e^{i\beta (x)Y}$. A left-handed doublet transforms as ${\psi }_L\mapsto e^{i\beta Y_L}U{\psi }_L$, while a right-handed singlet transforms as ${\psi }_R\mapsto e^{i\beta Y_R}{\psi }_R$. The Higgs doublet transforms as $\varphi \mapsto e^{i\beta /2}U\varphi$.

The representations per fermion generation are:

There is no right-handed neutrino in the minimal Standard Model. If such a sterile singlet were added, its hypercharge would be zero. The two hypercharge columns give equivalent normalizations, not different physical assignments. This is the electroweak representation and hypercharge table.

Yukawa interactions use $\varphi$ for down-type quarks and charged leptons, but $\tilde{\varphi }=i{\sigma }^2{\varphi }^{\ast }$ for up-type quarks. The conjugate doublet transforms as $e^{-i\beta /2}U\tilde{\varphi }$, using the pseudoreality of the $SU(2)$ fundamental representation. For example, the hypercharge sums in the down, charged-lepton, and up terms areThe doublet indices contract between ${\bar{\psi }}_L$ and the scalar, and colour indices contract in the quark terms. This verifies gauge invariance.

The indices $i,j=1,2,3$ label fermion generations, specifying which left-handed family and right-handed family are coupled. They are not weak-isospin indices. Each fermion type has its own complex Yukawa matrix. After symmetry breaking these matrices determine masses; the mismatch of the two quark left-handed mass rotations produces the CKM matrix. Left fields are weak doublets, right fields are weak singlets, and the scalar is a hypercharge-$1/2$ doublet in the stated convention.