Codex @codex  0

Use units $\hslash =c=1$, the Minkowski metric ${\eta }_{ab}=\mathrm{diag}(1,-1,-1,-1)$, and the Fourier transform convention $f(x)=\int d^{4}p{e}^{-ip\cdot x}\tilde{f}(p)/(2\pi {)}^4$. Here gamma matrices satisfy $\{{\gamma }^a,{\gamma }^b\}=2{\eta }^{ab}{I}_4$, and $\bar{\psi }={\psi }^{†}{\gamma }^0$ is the Dirac adjoint. A single Dirac field describes both Electrons and Positrons; these are the particle and antiparticle sectors of that field.

The free Maxwell Lagrangian and Dirac action, with a linear covariant gauge condition for the photon, giveBefore gauge fixing, the Maxwell Lagrangian has zero modes in field theory along $A_a\mapsto A_a+{\partial }_a\lambda$, so its quadratic operator cannot be inverted on all potentials. In the linear Lorenz gauge, the Faddeev-Popov determinant is $\det \square$. It is independent of $A$ and may be absorbed into the normalization; the corresponding Faddeev-Popov ghost fields have no interacting vertices in this Abelian linear gauge. A residual gauge symmetry is removed by the specified boundary conditions.

The free generating functional isThe Dirac field variables and their sources are independent Grassmann fields in this path integral. They anticommute; replacing them by ordinary commuting fields would give the wrong statistics and the wrong functional determinant. The bosonic Gaussian functional integral contributes an inverse square root of a determinant, and the Grassmann Gaussian integral contributes a determinant. If $K_A$ is the photon quadratic operator and $K_D=i{\gamma }^a{\partial }_a-m$, completing the square givesIn the last expression $D_F=i{K}_A^{-1}$ and $S_F=i{K}_D^{-1}$ are the photon propagator and Dirac propagator, with Feynman i-epsilon prescription. The products include the appropriate spacetime integrals and index contractions. Functional derivatives with respect to $J$, and consistently ordered left or right Grassmann derivatives with respect to the fermionic sources, generate the time ordering of the corresponding fields. In Feynman gauge, $\alpha =1$, the momentum-space two-point functions areThe Feynman i-epsilon prescription specifies vacuum boundary conditions rather than an arbitrary inverse of the differential operator.

The covariant photon propagator uses four potential components. Its operator formalism counterpart is Gupta-Bleuler quantization: impose $({\partial }_a{A}^a{)}^{(+)}|\mathrm{phys}⟩=0$ and take the Gupta-Bleuler null-state quotient. This leaves a positive physical state space with two transverse photon polarization vectors. The temporal and longitudinal oscillator components occur in intermediate covariant expressions; the Ward identity removes their dependence from physical amplitudes. The free Dirac field uses the canonical anticommutation relations, producing the same fermionic signs as its Grassmann fields in the path integral.

The operator-path-integral equivalence can be seen directly with a regulator. Divide time into small intervals and insert complete sets of field-coordinate states for bosons, and resolutions in fermionic coherent states for fermions. The bosonic matrix elements produce the phase-space factor $\exp i\sum [\pi \Delta \varphi -H\Delta t]$; integrating out the quadratic canonical momentum produces the bosonic action. The fermionic coherent state overlaps produce the first-order term $i{\psi }^{†}\dot{\psi }$ and the Berezin integral measure. Multiplying the short-time kernels recovers the path integral. Projecting the remote endpoints onto the Fock vacuum with an infinitesimal damping selects the same Feynman propagators as the operator formalism. Field insertions become time-ordered products under this construction. Conversely, their quadratic generating functional obeys the canonical free-field equations and has precisely the oscillator two-point functions, so its higher free correlators agree by the Wick theorem. This establishes the equivalence for the regulated free theory and order by order in the perturbation series.

To couple the matter field electromagnetically, promote its global phase symmetry to the local transformationThe gauge covariant derivative $D_a={\partial }_a+ie{A}_a$ obeys $D_a^{\prime }{\psi }^{\prime }=e^{-ie\lambda }{D}_a\psi$. Replacing ${\partial }_a$ by $D_a$ in the Dirac action therefore gives the invariant matter densityThe electromagnetic field tensor is unchanged by the local transformation. Thus the unfixed quantum electrodynamics action is gauge-invariant. The Dirac current $j^a=\bar{\psi }{\gamma }^a\psi$ is conserved by the matter equations, and the Electron and Positron excitations carry opposite charges. The added gauge fixing density selects a representative and is not itself invariant under arbitrary local transformations; it does not change gauge-invariant observables. At a free fermion vertex,For external on-shell Dirac spinors the corresponding current contraction vanishes. The quantum extension is the Ward identity, which makes physical amplitudes insensitive to adding a multiple of the photon momentum to its polarization vector. A gauge-compatible regularization preserves this vector-current identity.

With ${\mathcal{L}}_{\mathrm{int}}=-e\bar{\psi }{\gamma }^a\psi A_a$, expand $\exp (i\int {\mathcal{L}}_{\mathrm{int}})$ in powers of $e$. A term of order $n$ contains $n$ spacetime integrations and $1/n!$. Applying the Wick theorem pairs the free fields: an $A$-$A$ Wick contraction supplies a photon propagator, and a $\psi$-$\bar{\psi }$ Wick contraction supplies an oriented Dirac propagator. Each insertion supplies an interaction vertex. The permutations of contractions cancel the expansion factorials except for the Feynman-diagram symmetry factor. Interchanging Grassmann fields produces the fermionic sign, including a minus sign for every closed fermion loop. This is how Feynman diagrams arise from expectation values, rather than an extra dynamical assumption.

The normalized vacuum generating functional removes components with no external insertions. In the operator formalism, for an interacting-vacuum expectation value of an inserted product $\mathcal{O}$, the same cancellation appears asVacuum projection and the Feynman i-epsilon prescription are implicit. The linked-cluster theorem exponentiates all connected vacuum bubbles into the same factor in numerator and denominator, so it cancels. Normalize by $Z[0]$ to remove every vacuum component. This cancellation of vacuum bubbles still leaves products of disconnected diagrams that each contain external insertions. If only connected correlators are wanted, differentiate the connected generating functional $W=-i\log [Z[J,\bar{\eta },\eta ]/Z[0]]$.

The resulting momentum-space QED Feynman rules for the bare theory can be stated in Feynman gauge as follows:

The illustrated interaction vertex has incoming fermion momentum $p$, incoming photon momentum $k$, and outgoing fermion momentum $p^{\prime }=p+k$. Its solid-line arrows indicate fermion flow. The photon wavy line has no fermion-flow arrow. The propagators, the vertex $-ie{\gamma }^a$, momentum conservation, and the fermionic signs determine the perturbative amplitudes.

In a point-particle description of quantum gravity, interactions of a massless spin-two field occur at localized vertices of Feynman diagrams. The gravitational coupling has negative mass dimension for spacetime dimension greater than two, so short-distance loop integrations generate higher-derivative counterterms that cannot all be absorbed into the Einstein action. A string worldsheet instead joins incoming and outgoing strings by a smooth surface. In particular, a closed string splits or joins through a smooth pair-of-pants surface. There is no invariantly distinguished point at which every part of an extended string interacts. The choice of a time slicing can draw a joining point, but it is not a physical pointlike vertex. This removes the point-interaction geometry responsible for arbitrarily localized interactions and introduces the string length ${\ell }_s=\sqrt{{\alpha }^{\prime }}$.

This geometric explanation is supported by the organization of the Polyakov path integral. Introducing a worldsheet metric makes the Nambu–Goto action amenable to gauge fixing. In the Euclidean integral, metrics on a fixed oriented surface are divided by worldsheet diffeomorphisms and Weyl transformations. What remains is a Riemann surface with a conformal structure, together with finitely many worldsheet moduli and the positions of external string vertex operators. The determinants from fixing the metric are represented by Faddeev-Popov ghosts. Schematically, for $n$ closed-string external states,Here ${\mathcal{M}}_{g,n}$ is the moduli space of genus-$g$ punctured surfaces. The dilaton weights a surface by its Euler characteristic $\chi =2-2g$, giving $g_s^{-\chi }$ before the normalization of external states. Every additional handle multiplies the contribution by $g_s^2$. The string genus expansion therefore plays the role of the ordinary loop expansion, with the sphere giving tree level and the torus giving one loop. The critical dimension cancels the worldsheet Weyl anomaly, making this reduction to conformal geometry consistent.

The massless symmetric traceless closed-string state is a graviton, and its long-wavelength interactions reproduce gravity. At distances comparable with ${\ell }_s$, however, an infinite tower of excited states contributes. This is more than a single particle form factor: the same amplitude contains graviton exchange, massive higher-spin exchange and string-scale softness.

The dimensionless variables in the given Virasoro–Shapiro amplitude correspond to $s={\alpha }^{\prime }S/4$, $t={\alpha }^{\prime }T/4$, $u={\alpha }^{\prime }U/4$, where capital letters denote the usual dimensionful Mandelstam invariants. Its constraint $s+t+u=-4$ identifies the external particles as the closed bosonic tachyons, not four gravitons. Nevertheless its intermediate massless pole directly exhibits graviton exchange. The overall normalization will be suppressed below; no additional copy of the amplitude formula is needed.

First, the amplitude is symmetric under permutations of $s,t,u$, expressing crossing symmetry. The poles in any channel areFor generic $t$, the residue can be calculated with the Gamma function pole at $-n$ and its recurrence:The product is $1$ when $n=0$. Indeed, the pole factor contributes $(-1{)}^{n+1}/n!$, while the two remaining Gamma ratios contribute $(-1{)}^n(t+2{)}_n^2/n!$. A residue polynomial of degree $2n$ shows exchange of spins up to $2n$, with leading Regge trajectory $J=2+2s$. At $n=1$, the massless residue is $-(t+2{)}^2$, containing the spin-two exchange. By crossing, the massless $t$-channel pole isfor generic $s$. Thus at large $s$ the pole behaves as $s^2/t$, the characteristic energy dependence of graviton exchange. Its massless propagator produces the long-range gravitational interaction. At low energies the massive poles can instead be expanded into local higher-derivative corrections to the effective gravitational action.

At the string scale one must retain the whole tower. The Gamma reflection formula rewrites the amplitude exactly asAt large $s$ and fixed $t$, away from the resonance poles or with a specified complex continuation, the Gamma ratio scales as $s^{t+1}$. Hence the Regge limit scales as $s^{2+2t}$, up to the signature factor and the displayed $t$-dependent coefficient. In particular, a fixed nonzero momentum transfer changes the point-graviton power law by a factor $s^{2t}$: the graviton belongs to an entire trajectory rather than remaining an isolated spin-two exchange at arbitrarily high energy.

There is stronger suppression when the scattering angle is fixed. Put $t\simeq -as$, $u\simeq -(1-a)s$ with $0<a<1$. Applying Stirling formula to the Gamma functions, after the same treatment of real-axis poles, gives the leading smooth envelopeThe leading $s\log s$ terms cancel and the remaining entropy-like combination is negative. In physical units the exponent is $-{\alpha }^{\prime }Sf(a)/2$. This fixed-angle softness of the closed-string amplitude is absent for an elementary pointlike gravitational vertex. Equivalently, short-distance gravitational scattering probes additional string degrees of freedom. In a four-dimensional long-distance effective setting, a massive exchanged mode contributes a Yukawa-type correction proportional to $e^{-M_{n}r}/r$, with couplings and tensor factors depending on the external states. Such terms are suppressed for $r\gg {\ell }_s$ and become relevant for $r\sim {\ell }_s$. The amplitude establishes the spectrum and short-distance change; it does not by itself give a universal static potential for arbitrary sources. Its bosonic tachyon also signals an unstable vacuum, so it must not be treated as a complete stable model of a gravitational force.

Finally, the modular group prevents overcounting equivalent surfaces and reorganizes the loop ultraviolet region. A torus is $\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$ with $\mathrm{Im}\tau >0$. A change of lattice basis identifiesThus the one-loop integral uses a fundamental domain for $\mathrm{PSL}(2,\mathbb{Z})$, for exampleThe measure $d^2\tau /(\mathrm{Im}\tau {)}^2$ is invariant, and the physical integrand, including matter, ghosts and external insertions, must respect torus modular invariance. In this domain $\mathrm{Im}\tau \ge \sqrt{3}/2$. The region of arbitrarily small proper time that causes ultraviolet divergences in a particle loop is therefore not an independent part of the string integral; a modular transformation relates it to another description already represented in the domain.

This does not remove every possible divergence. The cusp $\mathrm{Im}\tau \to \infty$ is a long tube, and factorization through it can produce infrared divergences, massless tadpoles or the bosonic tachyon divergence. At higher genus, degenerating cycles similarly factorize into propagating intermediate string states. The essential improvement is the relation between worldsheet geometry, the full spectrum and the quotient by large diffeomorphisms; infrared and unstable-background problems remain separate questions. A short-distance expansion that keeps only the graviton discards precisely this structure.

Take a canonical Kähler potential $K={\Phi }^{†}\Phi$ and the Wess–Zumino model superpotentialThe F-term component givesEliminating the auxiliary field by $F=-\overline{W^{\prime }(\varphi )}$ gives the F-term scalar potential $V=|W^{\prime }(\varphi ){|}^2$. ConsequentlyThis is the supersymmetric relation between quartic and Yukawa couplings: there is one holomorphic parameter $g$, with the quartic coefficient $|g{|}^2$ and the Yukawa coupling $g$ in the displayed normalization. For real $g$ the quartic coefficient is the $g^2$ of the question. They cannot be varied independently while retaining this canonical supersymmetry.

The relation produces the supersymmetric cancellation of quadratic divergences. Bosonic and fermionic Feynman diagrams correcting a scalar mass have equal ultraviolet quadratic coefficients and opposite signs, so the mass is not driven directly to the ultraviolet cutoff scale. In a theory with soft supersymmetry breaking, residual corrections instead have the schematic sizewith coefficients and logarithms depending on the spectrum. Thus superpartner masses near the electroweak scale can stabilize the Higgs boson mass against a much larger cutoff, addressing the radiative part of the hierarchy problem. Heavy sparticles leave a corresponding residual sensitivity; the cancellation does not by itself explain every mass scale.

Forbidden by electric-charge conservation. The initial bottom quark has electric charge $-1/3$, while the displayed final electric charges sum to $2/3+1+0=5/3$. Therefore no Standard Model Feynman diagram, at tree level or any loop order, can realize this channel.

This is an intentionally disallowed process, not a transcription correction. The charge-conserving semileptonic bottom decay instead has an Electron and an antineutrino of Electron flavour, while a Positron accompanies the conjugate antiquark decay. Replacing the displayed particles silently would change the requested classification.

Take integer $n\ge 3$ and positive integer $d$, so the interaction is genuinely non-Gaussian. For a connected two-point Feynman diagram with $V$ interaction vertices and $I$ internal lines, counting half-edges and using the loop order givesAt one loop $I=V$, hence $(n-2)V=2$. There are only two possibilities: a two-interaction vertex bubble diagram for $n=3$, or a one-interaction vertex tadpole diagram for $n=4$. The latter is independent of external momentum and changes the mass, not the kinetic term.

For cubic scalar field theory, introduce a positive auxiliary mass $\mu$ if needed to separate infrared divergences from ultraviolet divergences. The one-loop self-energy in the quantum effective action is $-{\lambda }^2{B}_d(p)/2$, withThis Feynman parameter representation follows by shifting the loop momentum; it is directly valid for a translation-invariant regulator, or for the convergent subtracted integral. Differentiating at $p^2=0$ givesThus the coefficient of the kinetic term changes by ${\lambda }^2\int d^{d}q/[(2\pi {)}^d(q^2+{\mu }^2{)}^3]/6$.

Only the cubic interaction has a momentum-dependent one-loop two-point correction.There is a terminology qualification: if “wave-function renormalization” means a required ultraviolet counterterm, rather than a finite correction to the field normalization, the large-$q$ radial integral behaves as $\int dq{q}^{d-7}$, and the answer isIt is logarithmically divergent at $d=6$ and power divergent above $6$ with a momentum cutoff. Among interactions with a relevant coupling or marginal coupling, the divergent case is uniquely $(n,d)=(3,6)$. The massless low-dimensional amplitude needs an infrared divergence prescription; one cannot take a local expansion at zero momentum without one. The $n=1,2$ cases, if admitted, are a source and a Gaussian mass term and have no interaction-induced wave-function renormalization.

Use signed one-dimensional external momenta $k_1+k_2+k_3=0$, with nonzero $k_i$. Let $P_L(k)=P_L(-k)$ be the linear cosmological density power spectrum at the time of evaluation, normalized by $⟨{\delta }^{(1)}(k){\delta }^{(1)}(k^{\prime })⟩=2\pi {\delta }^{(D)}(k+k^{\prime })P_L(k)$. Using $P_L$ absorbs the factor $a^6$ multiplying a product of three initial power spectra. All kernels below are symmetrized one-dimensional cosmological density kernels.

The connected B222 contribution to the one-loop matter bispectrum has a triangular Feynman diagram: each second-order external field has two Gaussian inputs, joined pairwise to the other two vertices. The connected B411 contribution to the one-loop matter bispectrum has a fourth-order vertex joined to both linear external fields, with its remaining two inputs contracted into a loop at that vertex. External stubs indicate the measured momenta; blue internal lines represent linear power spectra.

There are $2^3=8$ connected Wick contractions in the triangle, givingFor a fixed fourth-order external field, attaching the two labelled linear fields gives $4\times 3=12$ choices; the remaining pair contracts uniquely. ThereforeThese are contributions to the reduced connected one-loop matter bispectrum, with its external momentum-conserving Dirac delta function removed. Disconnected contractions have a zero external momentum and are excluded.

For the ultraviolet behaviour, the three $F_2$ denominators multiply to a negative square. In factThus for $|q|\gg |k_i|$ and a smoothly varying high-momentum spectrum,Each second-order vertex exhibits the ultraviolet softness of the second-order density kernel, so this term contains six powers of external momentum.

The fourth-order kernel is even simpler:With a hard-loop cutoff $q_h<|q|<\Lambda$, defineThe B411 contribution to the one-loop matter bispectrum from this range is exactlyFor fixed momentum ratios, it has two external derivative powers multiplying two long-mode power spectra, rather than the six-derivative stochastic structure of $B_{222}$. It is the leading deterministic UV-sensitive contribution for external modes below the nonlinear wavenumber. This is a comparison of derivative orders and generic cutoff sensitivity; an arbitrary specially chosen spectrum or vanishing shape can change their numerical ranking. A convergent integral can still depend on nonperturbative short scales. For a high-$q$ power law $P_L(q)\propto |q{|}^n$, the $B_{411}$ integral diverges for $n\ge 1$, whereas the $B_{222}$ integral diverges only for $n\ge 5/3$, with logarithmic divergences at the thresholds.

At the field level, there are six ways to contract a hard pair within ${\delta }^{(4)}$. Comparing $6F_4$ with $F_2$ givesThe appropriate second-order density Laplacian counterterm is thereforeIndeed its contribution with two linear fields is $-2C(\Lambda ){\sum }_{\mathrm{cyclic}}{k}_3^2{F}_2{P}_L(k_1)P_L(k_2)$, which cancels the displayed hard-loop term. The counterterm sign follows from ${\partial }_x^2\leftrightarrow -k^2$. Its finite coefficient must be matched to short-scale dynamics, as in the effective field theory of large-scale structure. A term proportional to ${\partial }_x^2{\delta }^{(1)}$ similarly renormalizes a linear response, but a purely linear insertion cannot by itself cancel this $⟨{\delta }^{(4)}{\delta }^{(1)}{\delta }^{(1)}⟩$ shape, since an odd Gaussian three-point function vanishes.

The perturbative expansion of $Z[J]$ contains arbitrary Feynman diagrams, including disconnected products. The linked-cluster theorem givesbecause the factorials from repeated connected components reproduce the exponential series. Therefore $W[J]=-\hslash \log Z[J]$ is the sum of connected Feynman diagrams.

The Legendre transform removes diagrams that disconnect upon cutting one internal line. Equivalently, every connected diagram is a tree whose vertices are exact one-particle-irreducible vertices and whose edges are exact propagators. Thusand its functional derivatives are the one-particle-irreducible correlation functions. Algebraically, differentiating the Legendre relations givesso an exact quantum field theory propagator joining two proper vertices is precisely the inverse Hessian matrix needed to reconstruct connected diagrams.

Expand the quintic interaction after the slow-fast split. Its term with three slow fields and two fast fields isThe first term of the cumulant expansion therefore containsSince the coincident fast-mode propagator isthe lowest-order correction isIts Feynman diagram is one five-valent vertex with three external slow-field legs and the remaining two legs contracted into a tadpole diagram.

With Euclidean source convention $+\int d^{d}x{J}_a{\varphi }_a$, the generating functional isHere $J_a(x)$ is a nondynamical source. Its functional derivatives insert fields:This is why $Z$ “generates” correlation functions.

With the notation of the question, $W[J]=-\log Z[J]$ is the Euclidean connected generating functional; it is distinct from a Wilsonian effective action. Define the classical fieldThe quantum effective action is the Legendre transformwhere $J$ is eliminated in favor of $\Phi$. Its first derivative isso at zero source the quantum expectation value is a stationary point of $\Gamma$. Moreover,where $G_{ab}=⟨{\varphi }_a{\varphi }_b{⟩}_{J,\mathrm{conn}}$ is the exact connected correlation function. Thus the second derivative of $\Gamma$ is the inverse exact propagator.

Perturbatively, $Z[J]$ sums all Feynman diagrams, including disconnected ones. The exponential formula for combinatorial structures says that its logarithm selects connected Feynman diagrams, so $W$ sums connected diagrams with the sign dictated by the convention above. The Legendre transform removes diagrams that disconnect when one internal line is cut; consequently $\Gamma$ sums one-particle-irreducible Feynman diagrams. Equivalently, every connected diagram is a tree assembled from one-particle-irreducible vertices and full propagators, and the Legendre transform inverts that tree construction.

Now let $J_a^{\prime }=J_b{U}_{ba}$. ThenChanging variables to ${\varphi }_b^{\prime }=U_{ba}{\varphi }_a$ and using invariance of both the action and functional measure gives $Z[J^{\prime }]=Z[J]$, hence $W[J^{\prime }]=W[J]$. In the Legendre transform, the pairing obeysChanging the source variable and using the invariance of $W$ therefore givesThe symmetry of the classical action and measure is inherited by the full quantum effective action when it has no quantum anomaly.

In the displayed connected Feynman diagram, each sextic vertex carries two slow external legs and the four remaining legs at each vertex are paired across the vertices. The second cumulant expansion has a factor $-1/2$, the slow legs can be selected in ${\left(\genfrac{}{}{0ex}{}{6}{2}\right)}^2$ ways, and the four cross-contractions can be paired in $4!$ ways. The coefficient is thereforeWriting ${\mathcal{S}}_{\zeta }=\{q:\Lambda /\zeta <|q|<\Lambda \}$, the local zero-external-momentum contribution iswith the integral restricted further so that $q_1+q_2+q_3\in {\mathcal{S}}_{\zeta }$. The three independent internal momenta agree with the diagram's loop order $L=4-2+1=3$.

Classify the connected contractions by the numbers $(r,4-r)$ of external slow legs on the two sextic vertices and by the number $c$ of fast propagators joining them. Besides the displayed $(2,2;c=4)$ graph, the distinct topologies are:

Exchanging the two vertices gives no new topology. All four are connected Feynman diagrams selected by the logarithm in the cumulant expansion. With an ideal sharp momentum shell and a projection at exactly zero external momentum, the single joining line in the last topology cannot carry shell momentum, so that topology gives zero to the local quartic coupling; it is still the remaining formal connected contraction.

The gluon two-point function through one loop contains the tree propagator and these one-particle-irreducible insertions: a gluon loop with two three-gluon vertices, a gluon tadpole with one four-gluon vertex, a Faddeev-Popov ghost field loop, a fermion loop, and the gluon two-point counterterm. Gauge fixing is required before these Feynman diagrams and the propagator are defined.

Only the fermion loop changes when the fermion representation changes. Its two gauge vertices contain the representation matrices $T_R^a$, and its color factor isWith the conventional normalization, $T(F)=1/2$ for the fundamental representation and $T(\mathrm{adj})=C_A=N$ for the adjoint representation of $SU(N)$. The momentum and spinor integral is otherwise the same, apart from the number and type of fermion species; the pure-gluon and ghost diagrams are unchanged.

The Quantum effective action contains the following diagrams. For ${\Gamma }_2$, there are the tree inverse propagator, the one-loop tadpole at order $\lambda$, and at order ${\lambda }^2$ the two-loop sunset and the two-loop tadpole with a tadpole insertion. For ${\Gamma }_4$, there are the tree vertex $-i\lambda$ and, at order ${\lambda }^2$, the three one-loop bubble diagrams in the $s$, $t$, and $u$ channels. Disconnected and one-particle-reducible graphs do not occur in the quantum effective action.

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 $W^{\pm }$.

An interaction depending on the field can be represented by substituting $(1/i)\delta /\delta J$ into its action and acting on the free generating functional. Expanding that exponential produces Feynman diagrams, contractions and vertex weights. Normalize by the resulting value at $J=0$ to cancel vacuum diagrams.

A radiative correction is a quantum correction beyond the classical or tree approximation, represented in perturbative quantum field theory by Feynman diagrams of positive loop order. Such corrections can alter a scalar potential and its derivatives; symmetries and renormalization constrain which changes are allowed.

The Gaussian contraction used while integrating out a momentum shell. Its Fourier support lies entirely in that shell. In particular, convolution with a purely low-momentum field is zero. This distinguishes shell Feynman diagrams from unrestricted loop diagrams.

For a connected six-external-leg Feynman diagram made from quartic vertices, $4V=6+2I$ and $L=I-V+1$ give $V=L+2$, $I=2L+1$. A tree therefore has two vertices and one internal line; partitioning six labelled external legs into two triples gives ten tree channels. A one-loop triangle has three vertices, three internal lines and two external legs at each vertex. With external labels fixed, that triangle has symmetry factor one. Other one-loop topologies also contribute to the full amplitude.

A supergraph is a Feynman diagram formulated with superfield propagators and vertices in superspace. Performing its spinor-derivative algebra organizes perturbative corrections into superspace integration terms and provides a proof of the local superpotential non-renormalization theorem.

A triangle Feynman diagram has a loop of three propagators joining three vertices. In six-dimensional cubic scalar field theory it gives a one-loop three-point proper vertex. Attaching a cubic tree vertex makes a four-point connected diagram, but the connecting propagator is a bridge, so the latter is a one-particle-reducible Feynman diagram.