R-parity ($R_p$)

R-parity is the multiplicative quantum number $R_p=(-1{)}^{3(B-L)+2s}$. Standard Model particles are even and their superpartners are odd.

Matter parity assigns odd parity to the quark and lepton chiral superfields and even parity to the Higgs chiral doublets and gauge superfields. It is related to component R-parity by $R_p=P_M(-1{)}^{2s}$. The Lorentz-invariant action contains an even number of fermions, so conserved matter parity is equivalent to conserved R-parity. Products of four matter superfields can be even while violating baryon number; matter parity alone does not exclude higher-dimensional proton-decay operators.

Matter parity assigns $-1$ to every matter chiral superfield, including a singlet $N$ describing a conjugate right-handed neutrino, and $+1$ to Higgs chiral superfields. Thus the quadratic superpotential term $\frac{1}{2}{M}_{ij}{N}_i{N}_j$ is even, although it changes lepton number by two. The higher-degree operators $QQQL$ and $(L{H}_u{)}^2$ are also even and violate baryon number or lepton number. Exact R-parity therefore forbids the renormalizable trilinear violations $LLE$, $LQD$, $UDD$, $N{H}_d{H}_u$, $NNN$, but does not imply conservation of these continuous charges in every allowed interaction.

R-parity violation permits interactions containing an odd number of superpartners. Renormalizable R-parity-violating interactions can violate lepton number or baryon number, and allowing both generically induces rapid proton decay.

Simultaneous trilinear baryon number and lepton number violation permits a squark to connect the two interactions and induce a dimension-six proton-decay operator. Its coefficient is of order ${\lambda }^{\prime }{\lambda }^{\prime \prime }/m_{\tilde{q}}^2$. A dimensional decay-width estimate then gives ${\Gamma }_p\sim |{\lambda }^{\prime }{\lambda }^{\prime \prime }{|}^2{m}_p^5/m_{\tilde{q}}^4$, up to phase space and hadronic matrix elements. The flavor antisymmetry of $U^c{D}^c{D}^c$ matters: the two down-type flavor indices must differ. For example, ${\lambda }_{112}^{\prime \prime }$ and ${\lambda }_{112}^{\prime }$ allow strange-squark exchange in $p\to e^{+}{\pi }^0$.

The R-parity violation vertices from $LQD$ and $UDD$ can both contain the same down-type squark. Write its linear couplings as $-\tilde{d}A-{\tilde{d}}^{\ast }{A}^{†}$ and its mass term as $-M^2|\tilde{d}{|}^2$. Eliminating it gives $A^{†}A/M^2$. The cross term is $({\lambda }^{\prime }{\lambda }^{\prime \prime \ast }/M^2)(\nu d_L-e{u}_L)(u^{c†}{d}^{c†})+h.c.$, with the color Levi-Civita symbol contracting the three quarks. It changes baryon number and lepton number together and supplies the dimension-six proton-decay operator for $p\to e^{+}{\pi }^0$. The two down-flavor indices in $UDD$ must differ, so strange- or bottom-squark exchange is allowed but identical down flavors vanish.

Exchanging a down-type squark of mass $M$ between $LQD$ and $UDD$ vertices produces a dimension-six effective operator with coefficient $C\sim {\lambda }^{\prime }{\lambda }^{\prime \prime }/M^2$. Dimensional analysis then gives a proton decay width ${\Gamma }_p=c_h|{\lambda }^{\prime }{\lambda }^{\prime \prime }{|}^2{m}_p^5/M^4$, with a dimensionless factor $c_h$ containing hadronic matrix elements and phase space. A lifetime bound ${\tau }_p>{\tau }_{\min }$ implies $|{\lambda }^{\prime }{\lambda }^{\prime \prime }|<M^2/\sqrt{c_h{\tau }_{\min }{m}_p^5}$. The bound is on the relevant flavour product and depends on the exchanged mass; dimensional analysis alone does not determine $c_h$.