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The bino is the spin-one-half gaugino of the hypercharge $U(1{)}_Y$ gauge group. Its adjoint representation is the singlet $(1,1,0)$, so it has no hypercharge of its own. After electroweak symmetry breaking, it can mix with the neutral wino and higgsinos into neutralinos.

For a Higgs doublet with hypercharge $1/2$ and vacuum expectation value $(0,v/\sqrt{2}{)}^T$, its gauge-covariant kinetic term gives the neutral gauge-boson mass matrixin the $(W^3,B)$ basis. Its eigenvalues are zero and $(g^2+g^{\prime 2})v^2/4$, and the Weinberg angle rotation gives the photon and Z boson. The two charged electroweak gauge bosons have $m_W^2=g^2{v}^2/4$. The zero eigenvalue expresses the unbroken electric-charge gauge symmetry.

An electroweak scalar triplet transforms in the three-dimensional adjoint representation of $SU(2{)}_L$. A complex triplet of hypercharge one has components of electric charge two, one and zero, and its neutral component can couple to two left-handed lepton doublets.

The Standard Model lepton doublet $L=({\nu }_L,e_L{)}^T$ has hypercharge $-1/2$, the electron singlet $e_R$ has $-1$, and the Higgs doublet has $1/2$. Therefore $-y_e\bar{L}\varphi e_R+h.c.$ is gauge-invariant even though a bare ${\bar{e}}_L{e}_R$ mass is not. The vacuum expectation value generates a Dirac mass $|y_e|v/\sqrt{2}$. In unitary gauge, the Higgs boson coupling is $-(m_e/v)h\bar{e}e$ after rephasing the electron.

A Higgs chiral doublet is a weak-doublet chiral superfield containing a Higgs complex scalar field, a higgsino and an auxiliary field. In the MSSM, $H_u$ has hypercharge $+1/2$ and $H_d$ has hypercharge $-1/2$. Their paired charges enable higgsino anomaly cancellation and the holomorphic need for two Higgs chiral doublets.

Set $\epsilon =i{\tau }_2$, whose Pauli matrix identity is $\epsilon {\tau }_i^{\ast }=-{\tau }_i\epsilon$. Exponentiation gives $\epsilon U^{\ast }=U\epsilon$ for $U\in SU(2)$. Hence complex conjugating a Higgs doublet and multiplying by $\epsilon$ restores the fundamental representation of $SU(2)$, while reversing hypercharge. A Higgs field of hypercharge $1/2$ yields a conjugate doublet of hypercharge $-1/2$. Its product ${\tilde{\varphi }}^{†}L$ with a lepton doublet $L$ of hypercharge $-1/2$ transforms in the trivial representation of the whole electroweak gauge group. At vacuum $(0,v{)}^T/\sqrt{2}$, this composite is $v{\nu }_L/\sqrt{2}$.

A single weak-doublet higgsino of hypercharge $+1/2$ contributes $1/4$ to the cubic hypercharge gauge anomaly, $1/4$ to the weak-squared hypercharge coefficient with fundamental index $1/2$, and $1$ to the mixed gauge-gravitational anomaly. A second Higgs chiral doublet of opposite hypercharge cancels all three through its higgsino. The pair also contributes an even number of weak doublets, avoiding a new Witten SU(2) anomaly. Higgs complex scalar fields do not cancel chiral fermion gauge anomalies.

A renormalizable MSSM superpotential is a holomorphic function of chiral superfields and cannot use their conjugates. Gauge-invariant up-type Yukawa couplings require $H_u$ of hypercharge $+1/2$, while down-type and charged-lepton Yukawa couplings require $H_d$ of hypercharge $-1/2$. A single doublet and its conjugate cannot play both roles in the superpotential. This is distinct from higgsino anomaly cancellation.

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.

With hypercharge normalized by $Q_{\mathrm{em}}=T_3+Y$, the renormalizable R-parity-conserving MSSM superpotential is $W=(y_u{)}_{ij}{U}_i^c(Q_j\cdot H_u)+(y_d{)}_{ij}{D}_i^c(Q_j\cdot H_d)+(y_e{)}_{ij}{E}_i^c(L_j\cdot H_d)+\mu H_u\cdot H_d$, with $A\cdot B={ϵ}_{ab}{A}^a{B}^b$. Additional renormalizable R-parity violation allows $\frac{1}{2}{\lambda }_{ijk}(L_i\cdot L_j)E_k^c+{\lambda }_{ijk}^{\prime }(L_i\cdot Q_j)D_k^c+\frac{1}{2}{\lambda }_{ijk}^{\prime \prime }{ϵ}_{abc}{U}_i^{c,a}{D}_j^{c,b}{D}_k^{c,c}+{\kappa }_i{L}_i\cdot H_u$. The two antisymmetries are ${\lambda }_{ijk}=-{\lambda }_{jik}$ and ${\lambda }_{ijk}^{\prime \prime }=-{\lambda }_{ikj}^{\prime \prime }$. The first, second and fourth additional terms violate lepton number; the third violates baryon number.

Keep the PDF's hypercharge signs, which are the negatives of the frequently used convention. Write $U,D,E,N$ for the displayed $U^R,D^R,E^R,N^R$ chiral superfields. Their gauge group representations are conjugate right-handed-matter representations; their Weyl spinor components are left-chiral charge conjugates. Thus assign baryon number $B(Q)=\frac{1}{3}$, $B(U)=B(D)=-\frac{1}{3}$, and lepton number $L(L)=1$, $L(E)=L(N)=-1$, with both Higgs fields neutral under $B,L$. The corresponding electromagnetic convention is $Q_{\mathrm{em}}=T_3-Y$.

Let $X\cdot Y={ϵ}_{ab}{X}^a{Y}^b$ contract the two weak doublets, and use the color Levi-Civita symbol for three color antitriplets. The most general homogeneous cubic superpotential with right-handed-neutrino superfields separates as $W_3=W_{B,L}+W_{/B\text{or}/L}$, whereandRepeated generation and color indices are summed. The four Yukawa coupling matrices are arbitrary. The weak and color contractions imply ${\lambda }_{ijk}=-{\lambda }_{jik}$ and ${\lambda }_{ijk}^{\prime \prime }=-{\lambda }_{ikj}^{\prime \prime }$, while ${\rho }_{ijk}$ is completely symmetric. The $\lambda ,{\lambda }^{\prime }$ terms have $\Delta L=1$; ${\lambda }^{\prime \prime }$ has $\Delta B=-1$; $\kappa$ and $\rho$ have respectively $\Delta L=-1,-3$. Their conjugate interactions have the opposite changes. All four Yukawa terms preserve both numbers; all five remaining types violate at least one. In particular, the neutral $N$ fields printed in the PDF require both $N{H}_1{H}_2$ and $NNN$ terms.

For completeness of this list, a cubic color gauge-invariant can contain a $Q$–$U$ or $Q$–$D$ pair and one colorless doublet, or three color antitriplets. Vanishing hypercharge gives $Q{H}_{2}U$, $Q{H}_{1}D$, $LQD$ and $UDD$. Three $Q$ doublets have no weak singlet. Among colorless terms, two doublets and a singlet give $LLE$, $L{H}_{1}E$, $L{H}_{2}N$ and $H_1{H}_{2}N$; the apparent $H_1{H}_{1}E$ contraction vanishes since the identical bosonic superfields commute. Three singlets yield only $NNN$. This also explains why no omitted cubic monomial is available. The word cubic excludes the linear $N$ term and the quadratic $H_1{H}_2$, $L_i{H}_2$, $N_i{N}_j$ terms; these are separate possible terms in the full renormalizable superpotential.

To demonstrate squark-mediated proton decay, take ${\lambda }_{11k}^{\prime }$ and ${\lambda }_{11k}^{\prime \prime }$ with $k=2$ or $3$. Identical down-flavor indices would make ${\lambda }_{111}^{\prime \prime }$ vanish. Both interactions couple to the same scalar ${\tilde{D}}_k$ of mass $M$. Suppressing overall convention-dependent vertex signs, their relevant two-component fermion bilinears areThe color index on $A$ is contracted with that on ${\tilde{D}}_k$; Lorentz spinor indices within each bilinear are contracted too. Together with $-M^2|{\tilde{D}}_k{|}^2$, its algebraic equation at momenta small compared with $M$ gives ${\tilde{D}}_k=-A^{†}/M^2$. Substitution gives $A^{†}A/M^2$, whose cross term is the four-fermion matching for squark-mediated proton decayIts electron term contains three quark fields of flavors $uud$ and a lepton, and changes baryon number and lepton number together. The Hermitian-conjugate terms describe both orientations of the process. The hadronic matrix element of the three-quark proton-decay operator between a proton and a neutral pion has the right quantum numbers and is not forbidden by a remaining symmetry. Hence the operator permits $p\to e^{+}{\pi }^0$: the lepton field creates the positron and the three-quark matrix element connects the proton to the pion. This supplies the mechanism rather than merely naming the two violating interactions.

Each fermion field has mass dimension $\frac{3}{2}$, so this effective operator has dimension six and coefficient $C_6={\lambda }^{\prime }{\lambda }^{\prime \prime }{}^{\ast }/M^2$. By dimensional analysis its contribution to the decay rate has the formHere $c_h$ collects phase space, running and the dimensionless hadronic factor; a dimensional estimate takes it of order unity rather than pretending to calculate a strong-interaction matrix element. Assuming this exchange dominates and there is no tuned cancellation, the supplied lifetime gives the proton-lifetime bound on a product of R-parity-violating couplingsThe last estimate uses $m_p\simeq 1\mathrm{GeV}$ and ${\tau }_{\min }=2.4\times 10^{64}{\mathrm{GeV}}^{-1}$. Keeping $m_p=0.938\mathrm{GeV}$ instead changes $6.5$ to about $7.6$. A numerical limit independent of the squark mass or hadronic factor is not determined by the supplied data.

Finally, matter parity is $P_M=(-1{)}^{3(B-L)}$: all matter chiral superfields, including $N$, are odd, and the Higgs chiral superfields are even. R-parity of a component field is $R_p=P_M(-1{)}^{2s}$. In a superpotential, the spin factor is irrelevant to the allowed monomials: each of $Q{H}_{2}U,Q{H}_{1}D,L{H}_{1}E,L{H}_{2}N$ has two odd matter factors and is even. Each of $LLE,LQD,UDD,NNN$ has three odd matter factors, and $N{H}_1{H}_2$ has one, so all are odd and forbidden. Higgs vacuum expectation values are even; the allowed Yukawa interactions therefore still generate ordinary fermion masses.

R-parity forbids all the baryon/lepton-number-violating cubic terms above, not every possible violating operator. In particular, $\frac{1}{2}{M}_{ij}{N}_i{N}_j$ is even under matter parity but changes lepton number by two: matter parity allows Majorana neutrino masses. Thus the literal unrestricted claim would be false for these very fields; its valid cubic interpretation gives precisely the requested exclusion while preserving the Yukawa coupling mass terms.

Use $Q=T_3+Y$ and $T_a={\tau }_a/2$. For each generation $r=e,\mu ,\tau$, the leptons areBoth are color singlets. The conjugate antilepton fields transform in the conjugate representations and have opposite hypercharge; they are already included by the conjugate fields in the Lagrangian. The two components of $L_r$ have electric charges $0,-1$, and $R_r$ has charge $-1$. There is no right-handed neutrino representation in this field content. These assignments use the electroweak representation and hypercharge table in the same hypercharge normalization as the scalar doublet.

For a doublet of hypercharge $Y$, defineThe $SU(2)$ term is absent for a singlet. Under the local transformation $\psi \mapsto e^{iY\omega }U\psi$, the gauge fields transform so that $D_{\mu }\psi \mapsto e^{iY\omega }U{D}_{\mu }\psi$. Consequently the required gauge-covariant kinetic terms areThe fields of definite chirality are defined with $P_{L,R}=(1\mp {\gamma }_5)/2$. Since $\bar{L}$ has hypercharge $+1/2$, the Yukawa interactionis invariant: its hypercharges add to zero, and $\bar{L}\varphi$ contracts the two doublet indices. A bare charged-lepton mass violates these assignments. No renormalizable neutrino Yukawa term is available without a right-handed neutrino. A gauge-invariant scalar potential depending on ${\varphi }^{†}\varphi$ can select the nonzero vacuum, without altering these kinetic and Yukawa constructions.

There is an explicit local unitary gauge transformation whenever $r_{\varphi }=(|{\varphi }_1{|}^2+|{\varphi }_2{|}^2{)}^{1/2}\ne 0$:Thus, with the real field $\rho =\sqrt{2}{r}_{\varphi }-v$,This proves the polar coordinates for the electroweak Higgs doublet constructively. The gauge choice is local about the nonzero vacuum; it need not be nonsingular at zeros of the doublet.

The three identical-generation kinetic terms permit unitary flavor changes. A singular value decomposition gives $U_L^{†}{Y}_{\ell }{U}_R=\mathrm{diag}(y_e,y_{\mu },y_{\tau })$ with $y_r\ge 0$. Rotate both components of each left doublet together, and the right singlets independently. In this basis,The left/right cross terms combine into this Dirac mass. Since the neutrinos have no mass matrix, their simultaneous left rotation preserves a diagonal charged current. This is charged-lepton Yukawa matrix diagonalization; the neutrinos remain massless in the stated renormalizable model.

The scalar kinetic term givesIntroduce $s_W=g^{\prime }/\sqrt{g^2+g^{\prime 2}}$ and $c_W=g/\sqrt{g^2+g^{\prime 2}}$. The combinations $W^{+}=(A_1-i{A}_2)/\sqrt{2}$, $W^{-}=(W^{+}{)}^{\ast }$, $Z=c_W{A}_3-s_{W}B$ and $A_{\gamma }=s_W{A}_3+c_{W}B$ therefore diagonalize the quadratic terms:The factor $1/2$ applies to the real neutral field, not to the complex charged field. The massless photon corresponds to the unbroken generator $T_3+Y$, which annihilates the vacuum. This gives the electroweak doublet gauge-boson mass matrix and its diagonal spectrum.