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Take all external momenta incoming, $q_1=p_1$, $q_2=p_2$, and $q_{2+r}=-k_r$. For a diagram whose middle leg is $a$ and whose two end pairs are $\{b,c\}$ and $\{d,e\}$, the Feynman rules giveEquivalently,The full connected tree amplitude is the sum over the fifteen choices described in part d, multiplied by the overall momentum-conserving delta function from part c.

The first three terms are the renormalized kinetic, mass, and quartic interaction terms. The remaining three are counterterms: $\delta Z$ gives wave-function renormalization, $\delta m^2$ renormalizes the mass, and $\delta \lambda$ renormalizes the four-point coupling. With all momenta entering a vertex, the momentum-space Feynman rules aretogether with momentum conservation at every vertex and an integral $\int d^{d}p/(2\pi {)}^d$ for each independent loop momentum.

With all momenta incoming, the Feynman rules areandEach vertex includes its momentum-conserving delta function. The combinatorial factors follow by assigning the identical fields to the differentiated and undifferentiated slots in every possible way.

The hierarchy problem is visible in the one-loop top-quark contribution to the Higgs boson mass parameter,In a supersymmetric theory the corresponding stop loop has the opposite quadratic term. If supersymmetry is softly broken, the remaining correction is roughlyThe two relevant one-loop Feynman diagrams are a Higgs two-point function with a closed top-quark loop and the analogous stop loop. Requiring the residual correction not to exceed the electroweak scale by many orders of magnitude gives the naturalness expectation $m_{\tilde{t}}$, and broadly other sparticle masses, at a few TeV or below. This is an order-of-magnitude argument rather than a mass bound.

R-parity iswhere $B$ is baryon number, $L$ is lepton number and $s$ is spin angular momentum. Standard Model particles have $R_p=+1$, while their superpartners have $R_p=-1$. The renormalizable R-parity-violating superpotential of the Minimal supersymmetric Standard Model iswith ${\lambda }_{ijk}=-{\lambda }_{jik}$ and ${\lambda }_{ijk}^{\prime \prime }=-{\lambda }_{ikj}^{\prime \prime }$.

Combining ${\lambda }_{i1k}^{\prime }{L}_i{Q}_1{D}_k^c$ with ${\lambda }_{11k}^{\prime \prime }{U}_1^c{D}_1^c{D}_k^c$ produces proton decay through exchange of a virtual right-handed down-type squark. The tree diagram has $u$ and $d$ entering the baryon-number-violating vertex, an internal ${\tilde{d}}_{Rk}$ line, and the lepton-number-violating vertex emitting ${\bar{\nu }}_i$ and a light quark. Hadronization givesThe invisible particle is an antineutrino of any family $i=1,2,3$, and electric-charge conservation requires a positively charged pion. The antisymmetry of ${\lambda }_{11k}^{\prime \prime }$ requires $k=2$ or $3$.

Integrating out a squark of mass $m_{\tilde{q}}$ gives a dimension-six interaction with coefficient $C\sim {\lambda }^{\prime }{\lambda }^{\prime \prime }/m_{\tilde{q}}^2$. Ignoring order-one hadronic and phase-space factors,For $m_p\sim 1\mathrm{GeV}$ and $m_{\tilde{q}}\sim 10^4\mathrm{GeV}$, the factor before the couplings is about $10^{-18}$--$10^{-17}\mathrm{GeV}$. A lifetime above $10^{34}$ years is about $10^{65}{\mathrm{GeV}}^{-1}$, so ${\Gamma }_p\lesssim 10^{-65}\mathrm{GeV}$ andat order-of-magnitude accuracy. Hadronic matrix elements and flavour choices alter the numerical coefficient, not the conclusion that simultaneous baryon- and lepton-number violation is extraordinarily constrained.

The matter-parity factor $(-1{)}^{3(B-L)}$ is odd for every quark or lepton superfield and even for either Higgs superfield. Each term in $W_{\mathrm{RPV}}$ contains an odd number of matter superfields, so R-parity forbids all four types of term. Standard MSSM Yukawa interactions contain two matter superfields and remain allowed.

Finally, the requested four-point interaction contains a squark, an antisquark, a $W$ boson and a gluon. It comes from the squark gauge-covariant kinetic termThe cross term isbecause the colour and weak generators commute. With all fields incoming, its Feynman rule iswith the conventional $1/\sqrt{2}$ conversion when $W^I$ is replaced by $W^{\pm }$.

With all momenta directed into each vertex, the momentum-space Feynman rules are:

Each vertex also carries its momentum-conserving delta function; integrate every independent loop momentum and attach the appropriate external wavefunctions and photon polarization vectors.

In a canonical Wess–Zumino model with cubic superpotential $g{\Phi }^3/3$, auxiliary elimination gives a complex-scalar quartic interaction with two legs of each field type. With all momenta incoming and the displayed coefficient, its Feynman rule is $-i(2!2!)|g{|}^2$. For $\phi =(A+iB)/\sqrt{2}$, the same quartic scalar potential is $|g{|}^2(A^2+B^2{)}^2/4$, with real-field rules $-6i|g{|}^2$ for four identical legs and $-2i|g{|}^2$ for two of each. Factorials depend on how the coefficient is defined, not on a new physical coupling.

In unitary gauge, ${\varphi }^{†}\varphi -v^2/2=vH+H^2/2$, so this Higgs potential gives $M_H^2=2\kappa v^2$, a cubic coefficient $\kappa v$ and a quartic coefficient $\kappa /4$. The corresponding Feynman rules are $-3i{M}_H^2/v$ and $-3i{M}_H^2/v^2$. Writing the potential with an additional factor $1/2$ changes the relation between its named coupling and the Higgs boson mass; it does not change these mass-based rules.

For canonical Kähler potential and $W=m{\Phi }^2/2+g{\Phi }^3/3$, eliminating the auxiliary field gives $V=|m\varphi +g{\varphi }^2{|}^2$. The cubic interaction Lagrangian is $-m^{\ast }g{\varphi }^{\ast }{\varphi }^2-m{g}^{\ast }\varphi ({\varphi }^{\ast }{)}^2$. Differentiating with respect to the three external fields gives the all-incoming Feynman rules $V_{\varphi \varphi {\varphi }^{\ast }}=-2i{m}^{\ast }g$ and $V_{{\varphi }^{\ast }{\varphi }^{\ast }\varphi }=-2im{g}^{\ast }$; the factor two counts identical scalar legs. For real $m,g$ and $\varphi =(A+iB)/\sqrt{2}$, the cubic Lagrangian is $-mgA(A^2+B^2)/\sqrt{2}$, giving $V_{AAA}=-3\sqrt{2}img$ and $V_{ABB}=-\sqrt{2}img$.

For $m^2>0$, the zero-dimensional scalar field theory with Euclidean action $m^2{x}^2/2+\lambda x^4/4!$ has a Gaussian integral normalization $I(0)=\sqrt{2\pi /m^2}$. Its Feynman rules assign $1/m^2$ to each propagator, $-\lambda$ to each quartic vertex, and the reciprocal Feynman-diagram symmetry factor to each vacuum diagram. The integral itself includes disconnected vacuum diagrams; its logarithm selects connected ones. At second order the three possible numbers of edges joining the two vertices are zero, two and four, with Feynman-diagram symmetry factors $128,16,48$ respectively. Their weights add to $35{\lambda }^2/(384m^8)$.

The Wick contractions give the general coefficient $I(0)(-1{)}^v(4v-1)!!/[v!(4!{)}^v{m}^{4v}]$. Successive absolute coefficients have ratio $(4v+3)(4v+1)/[24(v+1)m^4]$, which tends to infinity. Thus this is an asymptotic expansion as $\lambda \downarrow 0$, rather than a convergent power series. For every fixed truncation the Taylor theorem remainder estimate for $e^{-u}$, $u\ge 0$, can be integrated against the Gaussian distribution to justify the corresponding remainder bound.