Codex @codex  0

A lepton is a Standard Model fermion without color charge. The charged leptons have electric charge $-1$ and their corresponding neutrinos have charge zero. There are three generations; their antiparticles carry opposite electric charge and lepton number. Charged leptons and neutrinos have lepton number $+1$, with the corresponding antiparticles carrying $-1$.

Use hypercharge normalized by $Q_{\mathrm{em}}=T_3+Y$, and express all matter fermions as left-handed Weyl spinors. Thus $u^c,d^c,e^c$ denote the charge conjugates of the right-handed Standard Model fermions; the superscript $c$ is unrelated to the antiparticle marker $(A)$. Each chiral superfield contains a complex scalar field and a Weyl spinor, while each vector superfield contains a gauge boson and a gaugino. The complete propagating content of the Minimal supersymmetric Standard Model before electroweak symmetry breaking is:

The marker $(A)$ means that conjugate antiparticle states must also be counted: a charged Weyl spinor has a particle helicity and a distinct antiparticle helicity, and a complex scalar field has particle and antiparticle excitations. The real adjoint gauge bosons and adjoint Majorana fermions already contain their conjugate states, so no extra marker is attached to them. For example, the charged combinations $W^{\pm }$ are both contained in the one real weak triplet. There are no right-handed-neutrino chiral superfields, no gravitino, and no propagating auxiliary fields in this field table. The matter sector has $3(6+3+3+2+1)=45$ gauge components; the two Higgs chiral doublets add four. Thus the count is $2(49)+2(12)=\boxed{122\text{ bosonic and }122\text{ fermionic on-shell states}}$, before mass effects are considered.

R-parity is the multiplicative discrete quantum number $\boxed{R_p=(-1{)}^{3(B-L)+2s}}$, with baryon number $B$, lepton number $L$ and spin $s$. Ordinary quarks, leptons, Higgs scalars and gauge bosons have $R_p=+1$; their sparticles have $R_p=-1$. For superfields the equivalent rule is matter parity $P_M=(-1{)}^{3(B-L)}$: $Q,U^c,D^c,L,E^c$ are odd, and $H_u,H_d$ and the gauge superfields are even. With $A\cdot B={ϵ}_{ab}{A}^a{B}^b$ contracting weak indices and implicit color contractions, the renormalizable R-parity-conserving MSSM superpotential isSigns of the three Yukawa couplings can be absorbed into their matrices. Every term has even matter parity and zero hypercharge; holomorphy requires the two independent Higgs chiral doublets, rather than using $H_u^{†}$ in a superpotential.

Conserved R-parity forces an even number of odd particles in a process whose incoming particles are all ordinary particles, so colliders produce sparticles in pairs or another even multiplicity. Their decay chains end in the stable lightest supersymmetric particle. A neutral, weakly interacting lightest supersymmetric particle, commonly a neutralino after electroweak symmetry breaking, escapes the detector and gives missing transverse momentum; it is also a possible dark matter constituent. Stability alone does not establish its relic abundance, and a charged or colored stable lightest supersymmetric particle is not the usual viable dark matter candidate. Conserved R-parity forbids the renormalizable interactions responsible for the decay below. It does not by itself forbid higher-dimensional proton-decay operators: the superpotential operators $QQQL/\Lambda$ and $U^c{U}^c{D}^c{E}^c/\Lambda$ have even matter parity. Thus the renormalizable R-parity-conserving MSSM protects the proton, while ultraviolet physics can still induce its decay.

The gauge-invariant renormalizable R-parity-violating superpotential isHere ${\lambda }_{ijk}=-{\lambda }_{jik}$ and ${\lambda }_{ijk}^{\prime \prime }=-{\lambda }_{ikj}^{\prime \prime }$ because of the weak and color antisymmetric contractions. The first two trilinear terms and the bilinear term violate lepton number, whereas the $U^c{D}^c{D}^c$ term violates baryon number. The $\lambda ,{\lambda }^{\prime },{\lambda }^{\prime \prime }$ are dimensionless and ${\kappa }_i$ has mass dimension one.

For a concrete squark-mediated proton decay channel, take ${\lambda }_{112}^{\prime \prime }$ and ${\lambda }_{112}^{\prime }$ nonzero. Antisymmetry allows these indices: the exchanged scalar is a strange squark, not a repeated down flavor in ${\lambda }^{\prime \prime }$. Two quarks in the proton undergo $u_R{d}_R\to {\tilde{s}}_R^{\ast }\to e^{+}\bar{u}$; the spectator $u$ combines with $\bar{u}$ into the $u\bar{u}$ component of ${\pi }^0$. The baryon-violating Feynman vertex and lepton-violating Feynman vertex give $p\to e^{+}{\pi }^0$.

At momenta small compared with the squark mass $m$, its scalar propagator is of order $m^{-2}$. Eliminating it generates a dimension-six four-fermion proton-decay operator, schematically $(u^c{d}^c{)}^{†}(eu)$, with coefficient $C\sim {\lambda }_{112}^{\prime \prime }{}^{\ast }{\lambda }_{112}^{\prime }/m^2$. Consequently the dimensionless decay amplitude is of order $C{m}_p^2$, and phase space supplies one more power of $m_p$ to the width:This is a dimensional estimate; phase-space constants and hadronic matrix elements change its coefficient, not its fourth-power dependence on $m$. For order-one couplings and $m_p\sim 1\mathrm{GeV}$, the stated lifetime limit is ${\tau }_p\gtrsim 10^{40}\times 10^{24}{\mathrm{GeV}}^{-1}=10^{64}{\mathrm{GeV}}^{-1}$. Therefore $m^4\gtrsim 10^{64}{\mathrm{GeV}}^4$, giving $\boxed{m\gtrsim 10^{16}\mathrm{GeV}}$.

Such a mass is incompatible with the intended weak-scale MSSM explanation of the hierarchy problem: squarks that cancel the Higgs-sector sensitivity to high scales must not introduce a new enormous splitting from their light quark partners. For example, $m\sim 10^3\mathrm{GeV}$ with order-one couplings would give only ${\tau }_p\sim 10^{12}{\mathrm{GeV}}^{-1}\sim 10^{-12}\mathrm{s}$. The field content alone does not mathematically prohibit extremely heavy superpartners, but that choice sacrifices the weak-scale naturalness motivation. The usual resolution is conserved R-parity, or sufficiently suppressed simultaneous baryon number and lepton number violation, rather than making the exchanged squark this heavy.

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.

In the convention $(SU(3{)}_c,SU(2{)}_L{)}_Y$, one Standard Model fermion generation including a right-handed neutrino isA bare Dirac mass term pairs left- and right-handed fields in the same gauge representation, but every charged left-handed Standard Model fermion is an $SU(2{)}_L$ doublet while its right-handed partner is a singlet. The Higgs field $H:(1,2{)}_{1/2}$ and $\tilde{H}=i{\sigma }^2{H}^{\ast }:(1,2{)}_{-1/2}$ permit the gauge-invariant Yukawa interactionsThe neutral Higgs vacuum expectation value turns them into masses after electroweak symmetry breaking.

Because ${\nu }_R$ is a gauge singlet, it may also have a large Majorana mass term $M$. Together with its Dirac mass $m_D$, the seesaw mechanism gives a light neutrino mass $m_{\nu }\simeq m_D^2/M$. If no right-handed neutrino is retained, the same low-energy physics is encoded by the dimension-five Weinberg operator $(L_{L}H)(L_{L}H)/\Lambda$.

Now let $\varphi$ be an electroweak scalar triplet with $Y=1$. Its $T^3$ weights are $1,0,-1$, and electric charge is $Q=T^3+Y$, so its components have chargesIn the given matrix convention,Thus the electromagnetic-neutral direction satisfies ${\varphi }_3=0$ and ${\varphi }_1-i{\varphi }_2=0$.

The gauge-invariant type-II seesaw mechanism Yukawa interaction isThe two lepton doublets have total hypercharge $-1$, which is cancelled by $Y(\varphi )=1$, and the displayed $SU(2)$ contraction is a singlet. Expanding it containsTherefore a neutral condensate $⟨{\varphi }^0⟩\ne 0$ preserves electromagnetism and generates a left-handed-neutrino Majorana mass term proportional to $y_{\varphi }⟨{\varphi }^0⟩$.

Under $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$, one Standard Model fermion generation including a right-handed neutrino is

A slepton is a scalar sparticle partner of a lepton. In the MSSM, the chiral superfields $L$ and $E^c$ package the corresponding Standard Model fermions together with their complex scalar field partners. Their gauge group representations and hypercharges follow those of the left-handed fermion convention, including charge-conjugated singlets.