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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 the printed hypercharge convention; reversing every hypercharge relative to another common convention does not change the gauge invariants. Write $U,D,E,N$ for the singlet chiral superfields labelled with a superscript $R$ in the paper, and define the weak-doublet contraction $A\cdot B={ϵ}_{ab}{A}^a{B}^b$. The most general homogeneous cubic superpotential with right-handed-neutrino superfields separates as $W_3=W_{B,L}+W_{\mathrm{viol}}$, whereandRepeated generation and color indices are summed. Because the chiral superfields commute while the gauge contractions are antisymmetric,The factors $1/2$ and $1/6$ remove the corresponding overcounting.

Here is a completeness check, not merely a list of possible terms. A color-singlet cubic can have no colored fields, a fundamental–antifundamental pair, or three fundamentals or three antifundamentals. A colored pair must be $QU$ or $QD$: the remaining weak doublet and hypercharge select respectively $H_2$ or $H_1,L$. Three antifundamentals allow only $UDD$ at zero hypercharge; three $Q$ doublets have no weak singlet. With no colored fields, weak invariance requires zero or two doublets. Two positive-hypercharge doublets require $E$, giving $L{H}_{1}E$ and $LLE$, while $H_1\cdot H_1=0$. Opposite-hypercharge doublets require $N$, giving $L{H}_{2}N$ and $H_1{H}_{2}N$. With no doublets, only $NNN$ has zero hypercharge. These are exactly the nine structures above.

For baryon number and lepton number, assignand zero charges to the Higgs fields. This is the conjugate-singlet convention, so the neutrino Yukawa coupling conserves lepton number. The first four terms conserve both charges. The $LLE$ and $LQD$ terms carry $L=1$, the $UDD$ term carries $B=-1$, the $N{H}_1{H}_2$ term carries $L=-1$, and $NNN$ carries $L=-3$. The gauge-singlet neutrino interactions must therefore be included in the violating sector.

If “cubic” instead means a polynomial of degree at most three, there are also the lower-degree gauge invariantsThe Higgs mass term conserves both charges; $t_i{N}_i$, $M_{ij}{N}_i{N}_j$ and ${\mu }_i{L}_i\cdot H_2$ violate lepton number. These additional terms distinguish the full renormalizable superpotential from its homogeneous cubic part.

To obtain squark-mediated proton decay, combine a lepton number-violating $LQD$ vertex with a baryon number-violating $UDD$ vertex. A nonzero example is ${\lambda }_{112}^{\prime }$ with ${\lambda }_{112}^{\prime \prime }$, exchanging the singlet strange quark squark of mass $M_{\tilde{s}}$. The antisymmetry forbids ${\lambda }_{111}^{\prime \prime }$, so using three first-generation singlets would not be a valid example. The relevant component vertices contain $e_L{u}_L{\tilde{s}}^c$ and $u^c{d}^c{\tilde{s}}^c$, with the required color contraction. Connecting one vertex to the complex conjugate of the other through the massive scalar propagator produces schematicallyParentheses denote Lorentz scalar two-spinor contractions, dotted for the right-handed pair and undotted for the left-handed pair. This dimension-six proton-decay operator contains the three quarks $uud$ and an electron field that can create a Positron. Its hadronic matrix element $⟨{\pi }^0|{ϵ}_{abc}(u_R^a{d}_R^b)u_L^c|p⟩$ gives the channel $p\to e^{+}{\pi }^0$. The internal strange flavor need not appear in the final state. The two interactions together violate both continuous charges, as required for this channel.

The coefficient of the dimension-six effective operator has mass dimension $-2$. Thus dimensional analysis giveswhere $c_h$ is a dimensionless hadronic and phase-space factor. Its detailed value cannot be fixed by dimensional reasoning. Applying the historical lower bound supplied in the paper gives the proton-lifetime bound on a product of R-parity-violating couplingsThe numerical coefficient is only the accuracy of the requested dimensional estimate; the squark mass was not fixed by the question.

Finally, R-parity is $R_p=(-1{)}^{3(B-L)+2s}$. At the superfield level use matter parity $P_M=(-1{)}^{3(B-L)}$: $Q,U,D,L,E,N$ are odd and $H_1,H_2$ are even. Each conserving Yukawa coupling contains two matter chiral superfields and one Higgs field, so it is even. Every violating homogeneous cubic above is odd: $LLE,LQD,UDD,NNN$ contain three matter fields, while $N{H}_1{H}_2$ contains one. R-parity forbids all the listed cubic violations and preserves the Yukawa terms that generate fermion masses. The Higgs bilinear is also even, whereas the lepton bilinear $L{H}_2$ and the singlet linear term are odd.

The final assertion must not be read as forbidding every possible violation of continuous baryon number or lepton number. In particular, matter parity allows Majorana neutrino masses: $N_i{N}_j$ is even but carries $L=-2$, a concrete counterexample once quadratic terms are included. Higher-dimensional operators such as $QQQL$ and $(L{H}_2{)}^2$ are likewise even and violate the continuous charges. The verified statement here concerns the homogeneous cubic sector, including the neutrino singlet terms.

Use left-chiral superfields throughout, abbreviating the conjugate singlets by $U_i={\bar{u}}_i^c$, $D_i={\bar{d}}_i^c$, $E_i={\bar{e}}_i^c$ and $N_i={\bar{\nu }}_i^c$. The printed hypercharge convention is the negative of the usual one, so $Q_{\mathrm{em}}=T_3-Y$; consequently $H_1$ is the down-type and $H_2$ the up-type Higgs doublet. Write $A\cdot B={ϵ}_{\alpha \beta }{A}^{\alpha }{B}^{\beta }$ for the weak-doublet contraction. Assign baryon number $B(Q)=1/3$, $B(U)=B(D)=-1/3$, and lepton number $L(L)=1$, $L(E)=L(N)=-1$; both Higgs fields have $B=L=0$.

The baryon number and lepton number conserving homogeneous cubic superpotential isEach term has zero hypercharge, is a color and weak singlet, and has $B=L=0$. Higgs expectation values give the usual quark and charged-lepton Yukawa interaction masses and, here, Dirac neutrino masses. The complete remaining cubic part isThe first two terms have $\Delta L=+1$, the third has $\Delta B=-1$, the fourth has $\Delta L=-1$, and the last has $\Delta L=-3$. Their Hermitian conjugates reverse these signs. Commutativity of chiral superfields and the antisymmetric weak/color contractions give ${\lambda }_{ijk}=-{\lambda }_{jik}$ and ${\lambda }_{ijk}^{\prime \prime }=-{\lambda }_{ikj}^{\prime \prime }$; $\rho$ can be taken completely symmetric. All other coupling tensors are unrestricted complex tensors.

To check completeness, a cubic color singlet containing a quark doublet and an antiquark singlet must contain a colorless weak doublet; the allowed hypercharges give $Q{H}_{2}U$, $Q{H}_{1}D$ and $QLD$. Three color-antitriplet singlets can form a singlet with ${ϵ}^{abc}$, but hypercharge permits only $UDD$. Three quark doublets cannot be a weak singlet. With no colored fields, an even number of weak doublets is necessary. Two doublets and a singlet give precisely $L{H}_{1}E$, $LLE$, $L{H}_{2}N$ and $H_1{H}_{2}N$; contractions of two identical Higgs superfields vanish. With no doublets the only neutral triple is $NNN$. Thus $W_3=W_{3,\mathrm{cons}}+W_{3,\mathrm{viol}}$ is the most general homogeneous cubic superpotential for the printed fields, including their singlet neutrinos.

If “cubic” is intended to mean a polynomial of degree at most three, also addThe $\mu$ term conserves both charges; the other nonconstant terms have respectively $\Delta L=1,-2,-1$. This distinction will matter for the printed claim about R-parity.

For squark-mediated proton decay, choose ${\lambda }_{11k}^{\prime }$ and ${\lambda }_{11k}^{\prime \prime }$ with $k=2$ or $3$. The same down-type singlet squark can appear in both vertices; $k=1$ would give ${\lambda }_{111}^{\prime \prime }=0$ by antisymmetry. The first term includes an electron/up-quark/squark vertex, and the second an up-quark/down-quark/squark vertex. Exchanging a squark of mass $M$ and expanding its propagator at momenta much smaller than $M$ produces the dimension-six effective operatorThe left- and right-handed pairs are separately contracted spinor bilinears. This operator converts an up/down quark pair into a positron and an antiquark; the spectator up quark can combine with the antiquark into the neutral pion. Equivalently its hadronic matrix element between a proton and a pion is permitted, so it induces $p\to e^{+}{\pi }^0$. Having both the baryon number and lepton number violating vertices is what enables this channel.

Since $C$ has mass dimension $-2$, dimensional analysis givesThe dimensionless $c_h$ includes the hadronic matrix element and two-body phase space; the pion mass changes this factor rather than the leading dimensions. Dimensional grounds alone do not give a precise hadronic coefficient. Using the lifetime supplied in the paper, rather than substituting a different bound, gives $\Gamma <1/(2.4\times 10^{64})\mathrm{GeV}=4.17\times 10^{-65}\mathrm{GeV}$ and henceFor an illustrative $100\mathrm{GeV}$ squark this is about $6.5\times 10^{-29}{c}_h^{-1/2}$ when $m_p$ is rounded to $1\mathrm{GeV}$. The bound constrains the relevant flavour product and depends on the exchanged mass; separate smallness of each coupling does not follow without an additional assumption.

Define matter parity and R-parity byEvery matter superfield $Q,U,D,L,E,N$ has $P_M=-1$, whereas $H_1,H_2$ and the vector superfields have $P_M=+1$. For ordinary quarks and antiquarks, $3(B-L)=\pm 1$ is odd and $2s=1$, so $R_p=+1$. For charged leptons, neutrinos and their antiparticles, $3(B-L)=\mp 3$ is odd and $2s=1$, again giving $R_p=+1$; the singlet neutrino fermion obeys the same rule. Gluons, the photon, $W$ and $Z$ have $B=L=0$ and spin one, hence $R_p=+1$; Higgs bosons have $B=L=s=0$ and also $R_p=+1$. Thus every Standard Model particle is R-even, including the corresponding antiparticles. Changing spin by one half within a supermultiplet reverses $R_p$, so squarks, sleptons, gauginos and Higgsinos are R-odd.

Each conserving cubic term contains two R-odd matter superfields and one even Higgs superfield, so it is even under matter parity. Each term in $W_{3,\mathrm{viol}}$ has an odd number of matter superfields: three for $LLE,LQD,UDD,NNN$, and one for $N{H}_1{H}_2$. All are therefore forbidden. At the component level the two fermion spin factors in a Yukawa interaction multiply to $+1$, so the same selection rule follows directly from R-parity. This verifies all cubic baryon/lepton-number violating terms are forbidden while the Yukawa fermion mass terms survive.

The blanket assertion needs a qualification beyond homogeneous cubic terms. The Higgs $\mu$ term is even; $L{H}_2$ and $N$ are odd and forbidden. But $N_i{N}_j$ is even and permits a Majorana neutrino mass despite violating lepton number by two. Likewise the higher-dimensional operators $QQQL$ and $(L{H}_2{)}^2$ are even. Consequently R-parity alone does not enforce all-order conservation of baryon number and lepton number, or absolute proton stability; the stated precise exclusion holds for the violating cubic interactions above.

Two physical implications follow from multiplicative conservation of R-parity. Superpartners are produced in even numbers from an initial state consisting only of Standard Model particles, leading to pair production at the lowest threshold and parity-conserving cascades. The lightest supersymmetric particle is stable, since one R-odd particle cannot decay exclusively into R-even particles and no lighter R-odd state is available. A neutral weakly interacting lightest supersymmetric particle can therefore be a dark-matter candidate and produce missing-energy signatures; neutrality and a suitable abundance require further model assumptions.

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 $c_h$.