= Solution
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:
\Table[
|| Sector
|| Component field
|| $SU(3)_c\times SU(2)_L\times U(1)_Y$ representation
|| Spin
|| Families or copies
| $Q_i$
| $(u_{Li},d_{Li})^{(A)}$, quark <Weyl spinors>
| $(\mathbf3,\mathbf2,+1/6)$
| $1/2$
| $3$
| $Q_i$
| $(\widetilde u_{Li},\widetilde d_{Li})^{(A)}$, <squarks>
| $(\mathbf3,\mathbf2,+1/6)$
| $0$
| $3$
| $U_i^c$
| $(u_i^c)^{(A)}$, quark <Weyl spinor>
| $(\overline{\mathbf3},\mathbf1,-2/3)$
| $1/2$
| $3$
| $U_i^c$
| $(\widetilde u_i^c)^{(A)}$, <squark>
| $(\overline{\mathbf3},\mathbf1,-2/3)$
| $0$
| $3$
| $D_i^c$
| $(d_i^c)^{(A)}$, quark <Weyl spinor>
| $(\overline{\mathbf3},\mathbf1,+1/3)$
| $1/2$
| $3$
| $D_i^c$
| $(\widetilde d_i^c)^{(A)}$, <squark>
| $(\overline{\mathbf3},\mathbf1,+1/3)$
| $0$
| $3$
| $L_i$
| $(\nu_{Li},e_{Li})^{(A)}$, lepton <Weyl spinors>
| $(\mathbf1,\mathbf2,-1/2)$
| $1/2$
| $3$
| $L_i$
| $(\widetilde\nu_{Li},\widetilde e_{Li})^{(A)}$, <sleptons>
| $(\mathbf1,\mathbf2,-1/2)$
| $0$
| $3$
| $E_i^c$
| $(e_i^c)^{(A)}$, lepton <Weyl spinor>
| $(\mathbf1,\mathbf1,+1)$
| $1/2$
| $3$
| $E_i^c$
| $(\widetilde e_i^c)^{(A)}$, <slepton>
| $(\mathbf1,\mathbf1,+1)$
| $0$
| $3$
| $H_u$
| $(H_u^+,H_u^0)^{(A)}$, Higgs <complex scalar fields>
| $(\mathbf1,\mathbf2,+1/2)$
| $0$
| $1$
| $H_u$
| $(\widetilde H_u^+,\widetilde H_u^0)^{(A)}$, <higgsinos>
| $(\mathbf1,\mathbf2,+1/2)$
| $1/2$
| $1$
| $H_d$
| $(H_d^0,H_d^-)^{(A)}$, Higgs <complex scalar fields>
| $(\mathbf1,\mathbf2,-1/2)$
| $0$
| $1$
| $H_d$
| $(\widetilde H_d^0,\widetilde H_d^-)^{(A)}$, <higgsinos>
| $(\mathbf1,\mathbf2,-1/2)$
| $1/2$
| $1$
| Color <vector multiplet>
| $G_\mu^a$, <gluons>
| $(\mathbf8,\mathbf1,0)$
| $1$
| $1$ adjoint, $a=1,\ldots,8$
| Color <vector multiplet>
| $\widetilde G^a$, <gluinos>
| $(\mathbf8,\mathbf1,0)$
| $1/2$
| $1$ adjoint, $a=1,\ldots,8$
| Weak <vector multiplet>
| $W_\mu^I$, weak <gauge bosons>
| $(\mathbf1,\mathbf3,0)$
| $1$
| $1$ adjoint, $I=1,2,3$
| Weak <vector multiplet>
| $\widetilde W^I$, <winos>
| $(\mathbf1,\mathbf3,0)$
| $1/2$
| $1$ adjoint, $I=1,2,3$
| Hypercharge <vector multiplet>
| $B_\mu$, hypercharge <gauge boson>
| $(\mathbf1,\mathbf1,0)$
| $1$
| $1$
| Hypercharge <vector multiplet>
| $\widetilde B$, <bino>
| $(\mathbf1,\mathbf1,0)$
| $1/2$
| $1$
]
{title=Propagating MSSM fields before electroweak symmetry breaking}
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 $A\mathbin{\cdot}B=\epsilon_{ab}A^aB^b$ contracting weak indices and implicit color contractions, the renormalizable <R-parity>-conserving <MSSM superpotential> is
$$
W_{\mathrm{RPC}}=(y_u)_{ij}U_i^c(Q_j\mathbin{\cdot}H_u)+(y_d)_{ij}D_i^c(Q_j\mathbin{\cdot}H_d)+(y_e)_{ij}E_i^c(L_j\mathbin{\cdot}H_d)+\mu H_u\mathbin{\cdot}H_d.
$$
Signs 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 $H_u^\dagger$ 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 $U^cU^cD^cE^c/\Lambda$ have even <matter parity>. Thus \b[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> is
$$
W_{\mathrm{RPV}}=\frac12\lambda_{ijk}(L_i\mathbin{\cdot}L_j)E_k^c+\lambda'_{ijk}(L_i\mathbin{\cdot}Q_j)D_k^c+\frac12\lambda''_{ijk}\epsilon_{abc}U_i^{c,a}D_j^{c,b}D_k^{c,c}+\kappa_iL_i\mathbin{\cdot}H_u.
$$
Here $\lambda_{ijk}=-\lambda_{jik}$ and $\lambda''_{ijk}=-\lambda''_{ikj}$ because of the weak and color antisymmetric contractions. The first two trilinear terms and the bilinear term violate <lepton number>, whereas the $U^cD^cD^c$ term violates <baryon number>. The $\lambda,\lambda',\lambda''$ are dimensionless and $\kappa_i$ has mass dimension one.
For a concrete <squark-mediated proton decay> channel, take $\lambda''_{112}$ and $\lambda'_{112}$ nonzero. Antisymmetry allows these indices: the exchanged scalar is a strange <squark>, not a repeated down flavor in $\lambda''$. Two quarks in the proton undergo $u_Rd_R\to\widetilde s_R^*\to e^+\bar u$; 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 $p\to e^+\pi^0$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/iii/paper-40-proton-decay.png]
{title=Proton decay through baryon- and lepton-number-violating strange-squark exchange}
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 $(u^cd^c)^\dagger(eu)$, with coefficient $C\sim\lambda''_{112}{}^*\lambda'_{112}/m^2$. Consequently the dimensionless decay amplitude is of order $Cm_p^2$, and phase space supplies one more power of $m_p$ to the width:
$$
\Gamma_p\sim|C|^2m_p^5\sim|\lambda'\lambda''|^2\frac{m_p^5}{m^4},\qquad \boxed{\tau_p\sim\frac{m^4}{|\lambda'\lambda''|^2m_p^5}}.
$$
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 $m_p\sim1\,\mathrm{GeV}$, the stated lifetime limit is $\tau_p\gtrsim10^{40}\times10^{24}\,\mathrm{GeV}^{-1}=10^{64}\,\mathrm{GeV}^{-1}$. Therefore $m^4\gtrsim10^{64}\,\mathrm{GeV}^4$, giving $\boxed{m\gtrsim10^{16}\,\mathrm{GeV}}$.
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, $m\sim10^3\,\mathrm{GeV}$ with order-one couplings would give only $\tau_p\sim10^{12}\,\mathrm{GeV}^{-1}\sim10^{-12}\,\mathrm{s}$. 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.
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