 Proton-lifetime bound on a product of R-parity-violating couplings

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 $c_h$.