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For $\Psi (p_1)\to \chi (p_2)+\varphi (k_1)+\varphi (k_2)$, the sole tree-level Feynman diagram has a Yukawa coupling interaction vertex producing an off-shell scalar field particle of four-momentum $q=p_1-p_2=k_1+k_2$, followed by its cubic scalar field interaction vertex. Its invariant decay amplitude isTwo successive Yukawa coupling interaction vertices would change the fermion species twice and leave a $\Psi$, so they cannot produce this final state. Interchanging the two identical outgoing bosons does not create another Feynman diagram; the cubic factorial is already included in the Feynman rule. The integrated decay width instead includes the identical final-state symmetry factor $1/2!$.

Let $L$ be the number of independent loop momenta in a connected Feynman diagram. Every scalar propagator falls as $k^{-2}$, and every loop momentum integration contributes three powers, so $\omega =3L-2e$. Counting line ends and using connectedness givesEliminating $L$ and $e$ yieldsThus increasing the sextic interaction vertex count does not worsen the superficial degree of divergence; adding a quartic interaction vertex improves it. Only finitely many external-line structures can have an overall ultraviolet divergence, even though arbitrarily high loop order is possible.

After subtracting ultraviolet subdivergences, locality and Lorentz invariance restrict the divergent counterterms to a constant vacuum energy density, a mass counterterm, a wave-function renormalization, and quartic and sextic field interaction terms. Odd powers are forbidden by the $\varphi \mapsto -\varphi$ Z2 symmetry. In particular, $n=2$ permits a constant and $p^2$, $n=4$ permits only a constant, and $n=6$ permits only a constant. This is perturbative renormalizability, not a guarantee that every superficially allowed divergence actually occurs or that every unrenormalized graph with $\omega <0$ lacks divergent subgraphs.

For the two-point function, the original Feynman diagrams below show both one-vertex graphs and every two-vertex topology with external momentum flowing between distinct interaction vertices. All lines represent the same real scalar field. The lower row labels the number of parallel connecting lines; a small closed line at a vertex is a tadpole diagram.

To include all momentum-dependent contributions to the two-point function, put one external line at each interaction vertex. Let $r$ be the number of lines connecting the vertices, and $t_A,t_B$ their numbers of tadpole diagrams. For vertex valences $v_A,v_B\in \{4,6\}$,The integer $r$ is odd, and $r=1$ gives a one-particle-reducible Feynman diagram and is excluded. Thus the complete list is

The coefficient of the mixed graph includes the reflected external attachment as well as its reciprocal Feynman-diagram symmetry factor. If both external lines attach to the same interaction vertex, no external momentum enters the connecting internal lines, and that Feynman diagram cannot contribute to wave-function renormalization.

For incoming external momentum $p$, assign the connecting lines momenta $k_1,\dots ,k_{r-1},p-{\sum }_j{k}_j$. DefineThe momentum space Feynman rules give, in the order listed,Each $C$ is a reciprocal Feynman-diagram symmetry factor, including any equivalent external attachments. There are respectively $2,3,4,4$ independent loop momenta.

If “contribute to wavefunction renormalization” means requiring a divergent kinetic counterterm after subtraction of all subdivergences, only the last topology qualifies. The first three still have nontrivial external momentum dependence and give finite contributions to a normalization condition; their raw divergent factors arise from mass counterterms and interaction vertex counterterms. Keeping all four here makes this distinction explicit. The ultraviolet divergence analysis in d establishes it directly.

With a single quartic interaction vertex, the two unused line ends close into one tadpole diagram. With a single sextic interaction vertex, the four unused ends pair into two tadpole diagrams. These are the only one-vertex one-particle irreducible Feynman diagrams for the two-point function.

Define the Minkowski Feynman integralsThe amputated two-point function contributions arewhere $C_4,C_6$ are the reciprocals of the corresponding Feynman-diagram symmetry factors. Both are independent of external momentum, and hence affect the mass counterterm rather than wave-function renormalization. No numerical Feynman-diagram symmetry factors are needed here.

There are three independent nonfactorized momentum integrals, namely a tadpole integral, the three-line sunset diagram integral, and the five-line banana integral. The six displayed Feynman diagrams involve six complete integral expressions:The distinction between independent building blocks and their products removes an ambiguity in counting “different momentum integrals”. For the strict divergent-wave-function renormalization interpretation of ii, the graphs in c reduce to $T,T^2,B_5$, comprising three full integral expressions and two independent nonfactorized building blocks.

Make a Wick rotation and use $m>0$ to separate ultraviolet divergences from infrared divergences. Write $D_E(k)=1/(k^2+m^2)$, and let $T,B_r(P)$ be the Euclidean versions of the integrals above, with their harmless overall factors of $i$ removed. A spherical ultraviolet cutoff givesThus the one-vertex quartic graph is linearly divergent, and the one-vertex sextic graph is quadratically divergent as $T_{\Lambda }^2$.

For the other two building blocks, the spatial Fourier transform is especially transparent. The three-dimensional Euclidean scalar propagator isThe convolution theorem converts the $r$ parallel lines into $G(x{)}^r$. Angular integration gives, with a short-distance cutoff $a={\Lambda }^{-1}$,This is a different convenient cutoff regularization for the connected block; power-divergent coefficients depend on the cutoff convention, whereas the conclusions about powers and the presence of a kinetic logarithm do not. The sinc function is understood to be $1$ at $P=0$.

For $r=3$, the integrand starts as ${\rho }^{-1}+O(1)$, soIts divergent part is independent of external momentum. In particular,The finite difference and its finite derivative show why the pure three-line sunset diagram does not need a divergent wave-function renormalization in three dimensions.

For $r=5$, the small-distance Taylor expansion isConsequentlyIn particular, the nonzero $P^2\log \Lambda$ term requires a genuine divergent wave-function renormalization, in addition to the mass counterterm. The coefficients here refer to the displayed position-space cutoff and have not included Feynman-diagram symmetry factors or bare couplings.

The comparison with the superficial degree of divergence isThus the actual leading power agrees with $\omega$ in every case. The additional logarithms in products come from ultraviolet subdivergences, which uniform power counting alone does not resolve.

For $T{B}_3$ and $T^2{B}_3$, differentiation with respect to $P^2$ removes the logarithmic divergence of $B_3$, but leaves divergent factors $T$ or $T^2$ multiplying a finite, nonpolynomial function of $P^2$. These are subdivergent contributions, not new nonlocal counterterms: subtracting each tadpole diagram subdivergence and the momentum-independent $B_3$ subdivergence removes them. For example, in a factorized subtraction scheme $T{B}_3\mapsto (T-T_{\mathrm{div}})(B_3-B_{3,\mathrm{div}})$, and likewise for $T^2{B}_3$. The remaining $B_5$ kinetic logarithm explains the strict interpretation in ii. A dimensional regularization scheme can suppress power divergences; it does not invalidate this cutoff-based comparison with $\omega$.

A dimensionless action in six spacetime dimensions requires the Lagrangian density to have mass dimension six. The kinetic term then fixes $2+2[\varphi ]=6$, and each field interaction term must have the same mass dimension. ThereforeEquivalently, in powers of a length unit $L$, $\varphi$ has units $L^{-2}$, $g_1$ has units $L^0$, and $g_2$ has units $L^2$. Thus literal length exponents would be $(-2,0,2)$; the $d$'s in the supplied superficial degree of divergence formula must instead mean mass dimensions, namely $(2,0,-2)$. This distinction resolves the wording's convention mismatch.

To check the convention independently, let $I,V_3,V_4,\ell$ count internal lines, cubic interaction vertices, quartic interaction vertices, and independent loop momenta. A connected Feynman diagram obeys $2I+n=3V_3+4V_4$ and $\ell =I-V_3-V_4+1$. Hence its superficial degree of divergence is $6\ell -2I=6-2n+2V_4$, agreeing with the supplied formula only with the mass dimensions above.

Set $g=g_1$ and $g_2=0$. The superficial degree of divergence is then $\omega =6-2n$, independently of the number of cubic interaction vertices. After subtracting ultraviolet subdivergences, the divergent part of a one-particle irreducible Feynman diagram is a local polynomial in external momenta of degree at most $\omega$. Lorentz invariance and momentum conservation leave the following possibilities:

Thus a finite set of counterterms closes under perturbation theory. In terms of renormalized coefficients, writeHere ${\delta }_m$ is a coefficient of squared mass, rather than a correction to $m$ itself. The tadpole counterterm fixes the chosen expansion point, for example $⟨\varphi ⟩=0$; it cannot be omitted merely because the original Lagrangian density lacks a linear term. The vacuum energy density term cancels out of normalized Green functions, but is required if the unnormalized vacuum functional is also to be renormalized. This establishes perturbative renormalizability; the unbounded scalar potential of a real cubic scalar field theory does not by itself define a stable nonperturbative vacuum state.

Use $\hslash =c=1$, the Minkowski metric $\eta =\mathrm{diag}(1,-1)$, and $p^2=(p^0{)}^2-(p^1{)}^2$. The momentum-space Feynman rules for the real scalar field areAt each interaction vertex impose two-dimensional momentum conservation; before removing the overall conservation factor this is $(2\pi {)}^2{\delta }^{(2)}({\sum }_j{p}_j)$. Integrate each independent loop momentum with $d^{2}k/(2\pi {)}^2$, and divide a Feynman diagram by its symmetry factor. External lines carry the same scalar propagator in an unamputated two-point correlation function; they are removed in an amputated scattering amplitude.

The quadratic action is inverted with the Feynman i-epsilon prescription to obtain the scalar propagator. Expanding the Dyson series gives $-i{g}_4/4!$ per insertion; the $4!$ attachments in the Wick theorem cancel the factorial. Fourier integration over the insertion point supplies the momentum conservation factor. Automorphisms that leave the labelled external attachments fixed account for the remaining symmetry factor. In two spacetime dimensions the mass dimensions are $[\varphi ]=0$ and $[g_4]=2$.

Label the incoming four-momenta $p_1,p_2$ and the outgoing ones $p_3,p_4$. Define the Mandelstam variables $s=(p_1+p_2{)}^2$, $t=(p_1-p_3{)}^2$, $u=(p_1-p_4{)}^2$. Two incoming $\varphi$ particles have conserved charge $+2$, so the only tree-level Feynman diagram is an $s$-channel $\Phi$ exchange, shown in panel (b) of b's figure. Each end converts two charge-one lines into one charge-two line. There are no $t$- or $u$-channel exchanges: an incoming and an outgoing particle at such an end would transfer zero charge, for which this theory has no cubic interaction vertex.The single Feynman diagram is already symmetric under exchanging the two outgoing particles. This does not add another copy of the scattering amplitude; the identical final-state symmetry factor enters the integrated scattering cross-section instead.

Let $I_b,I_f$ count internal scalar and fermion lines and $L$ count independent loops in a connected amputated Feynman diagram. At large loop momentum, the scalar propagator falls as $k^{-2}$ and the Dirac propagator falls as $k^{-1}$, including its numerator. Each loop integration is $d^{3}k$. The nonderivative interaction vertices add no powers. ThereforeCount endpoints of each kind of line and use the connected-graph loop order identity:Eliminating $L,I_b,I_f$ givesThe cancellation of $n_1,n_3$ reflects their dimensionless couplings; every insertion of the positive-dimensional $g_2$ improves the superficial degree of divergence by one.

The printed exact homogeneity is shorthand for ultraviolet power counting, not an identity for a massive integrated correlation function. For example, even the free amputated scalar two-point vertex ${\Gamma }^{(2)}(p)=p^2-M^2$ obeys ${\Gamma }^{(2)}(\lambda p)={\lambda }^2{p}^2-M^2$, which differs from ${\lambda }^2{\Gamma }^{(2)}(p)$ when $M\ne 0$. Logarithmic divergences, actual numerator cancellations and ultraviolet subdivergences also require separate treatment.

The corrected expression in d is a vacuum two-point function, or Wick contraction, of the free Dirac field. It equals the negative spinor transpose of the usual Dirac propagator with reversed spacetime arguments. It supplies the internal fermion lines in Feynman diagrams, rather than a position-space detection probability.

For $x^0>y^0$, $\psi (y)$ creates an antiparticle and $\bar{\psi }(x)$ annihilates it. The $v\bar{v}$ term describes positive-energy antiparticle propagation from $y$ to $x$. For $y^0>x^0$, $\bar{\psi }(x)$ creates a particle and $\psi (y)$ annihilates it. The $u\bar{u}$ term describes positive-energy particle propagation from $x$ to $y$, with the minus sign required by exchanging the odd operators in fermionic time ordering. Neither term describes a physical negative-energy excitation.

The Heaviside step functions select these chronological alternatives. The factors $e^{\mp ip\cdot z}$ are on-shell plane waves with $p^0=E_{\mathbf{p}}>0$, and $d{\Pi }_p$ is the positive-energy Lorentz-invariant phase-space measure. The sum over $s$ includes the two spin states. With standard relativistic spinor normalization, the fermion spin sums areThey encode the spinor structure and solve the corresponding on-shell Dirac equations. In particular, the corrected answer can also be writtenThe Feynman i-epsilon prescription specifies the temporal boundary conditions of the Green function; its positive- and negative-frequency residues reproduce the two terms and their relative minus sign. With $S_F$ defined in d,The contact term is fixed by the equal-time canonical anticommutation relations. Choosing a value for $\theta (0)$ does not change the unequal-time sign demonstrated above.

Finally, microcausality requires field anticommutators to vanish at spacelike separation; the time-ordered vacuum two-point function itself need not vanish there. These facts are compatible with relativistic causality. The plus-plus expression printed in the paper lacks the fermionic sign needed for this standard Dirac propagator interpretation.

The charged weak current changes one neutron down quark into an up quark:The other two valence quarks are spectators in this tree-level Feynman diagram. The intermediate W boson is highly virtual; the available neutron decay energy is far below its on-shell rest energy. Charge is conserved at each vertex, and the electron's lepton number is balanced by the electron antineutrino.

The reversed quark-level charged current is allowed as an interaction, but a free proton cannot decay as $p\to n+e^{+}+{\nu }_e$ because $m_p<m_n+m_e$. Four-momentum conservation leaves no accessible final-state Lorentz-invariant phase space. The absence of this free-proton beta-plus decay is therefore kinematic, not an absence of a $u\to d$ weak vertex. Binding-energy differences in a nucleus, or incoming energy in a collision, can change whether the corresponding conversion is possible. Exotic baryon-number-violating proton decay is a separate question and is not generated by this ordinary charged-current diagram.

The first-order Dyson series contains $-ig\int d^{4}w{\phi }^{†}\phi \varphi$. For an amputated three-point Feynman diagram, each of the three distinct field types contracts in exactly one way. Writing each external plane wave in momentum space producesThus the momentum-space Feynman rule, with all four-momenta incoming, isAfter stripping the conservation delta factor, it is simply $-ig$. There is no $3!$ combinatorial factor because the interaction has three distinct field slots. The charged lines carry opposite charge-flow orientations and the real heavy line is neutral.

Set $C_{\mu \nu }=A_{\mu }{B}_{\nu }-A_{\nu }{B}_{\mu }$, so $\mathcal{G}=G+ieC$. Expanding the minimally coupled Lagrangian density givesThus there is a derivative cubic $AB{B}^{\ast }$ interaction vertex of order $e$, and a quartic $AAB{B}^{\ast }$ interaction vertex of order $e^2$. There is no pure $B^4$ contact term in this minimal coupling theory, nor an Abelian three-photon interaction vertex.

The straight arrowed lines in the original schematic denote charged massive vector bosons; the wavy lines denote photons. For two identical positively charged vector bosons, the leading scattering amplitude has the order-$e^2$ $t$-channel photon-exchange Feynman diagram, together with its $u$-channel interchange required by identical bosons. A quartic $AAB{B}^{\ast }$ interaction vertex cannot directly attach four external massive vector bosons. For a particle-antiparticle incoming pair, the leading channels instead are $t$-channel exchange and $s$-channel photon annihilation. The shown arrows track charge flow; an outgoing charged particle corresponds to the conjugate field at a vertex.

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.

For Hermitian fundamental representation generators normalized by $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}/2$, the Abelian-shaped one-loop Feynman diagram has colour factor ${\sum }_b{T}^b{T}^a{T}^b=(C_F-C_A/2)T^a=-T^a/(2N)$. Expand ${\sum }_b[T^b,[T^b,T^a]]=C_A{T}^a$ to prove the identity. With oriented structure tensor $f^{abc}$, the three-gluon graph has $f^{abc}{T}^b{T}^c=i{C}_A{T}^a/2$ by antisymmetrizing the generator product. The factor $i$ and its sign must be combined consistently with the chosen three-gluon Feynman rule. For $SU(3)$, the two real coefficients are $-1/6$ and $3/2$.

Quantum field theory combines special relativity with quantum theory by treating fields as operator-valued distributions and particles as field excitations.

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.