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For external electrons on shell, factor out signed electric charge in the parity-preserving interaction vertex:The $u$ are Dirac spinors; ${\gamma }^{\mu }$ are gamma matrices. With on-shell renormalization scheme normalization $F_1(0)=1$, the Gordon identity gives $g_s/2=F_1(0)+F_2(0)$ and $a_e=F_2(0)$.^[ref]

This explains a subtlety: the Dirac equation already predicts a spin magnetic dipole moment with $F_2=0$. Rewriting its Dirac electromagnetic current with the Gordon identity produces a spin-dependent term even before any loop order correction. $F_2$ measures the additional response after that baseline is included. The limit $q\to 0$ is taken after extracting its coefficient; setting $q=0$ in the interaction vertex at the outset would hide it. This electromagnetic anomalous magnetic moment is not a failure of gauge invariance of the kind called a quantum anomaly.

For a massive real scalar field in two spacetime dimensions with a polynomial potential and no derivative interactions, every connected subgraph has superficial degree of divergence $\omega =2(1-V)$, where $V$ counts interaction vertices. The only divergent loop subgraphs with external legs have one vertex and consist of self-contractions. Normal ordering removes them. With $T_{\Lambda }={\int }_{|k|<\Lambda }{d}^{2}k/[(2\pi {)}^2(k^2+m^2)]$, the ordinary-field potential representing the normal-ordered interaction is $e^{-T_{\Lambda }{\partial }_{\varphi }^2/2}V(\varphi )$. The exponential terminates for a polynomial. After these subtractions all remaining loop subgraphs have negative superficial degree of divergence, so no additional divergent field-dependent counterterms occur. Constant vacuum energy subtractions and finite on-shell renormalization scheme conditions are separate.

The on-shell renormalization scheme fixes masses at physical propagator poles, fixes field normalizations by pole residues, and fixes couplings through amplitudes evaluated with external particles on their mass shells.

An on-shell renormalization scheme fixes the mass parameter at the physical pole mass and fixes the renormalized field by the chosen pole residue. Its counterterms include finite contributions needed to enforce those conditions. A coupling is likewise defined by a specified physical amplitude or mass-shell condition.

The minimal subtraction scheme removes only poles in the dimensional regulator. The modified minimal subtraction scheme, which is the overbarred scheme in the PDF, removes the accompanying universal combination $\log 4\pi -{\gamma }_E$ as well. In $D=4-ϵ$, the one-loop subtraction is proportional toA subtraction-scheme mass is a running parameter and generally differs from the physical mass. Finite redefinitions relate the schemes while leaving physical quantities unchanged. The overbar is essential for the stated finite formula in part (d).

For the Euclidean quartic scalar field theory, expansion of $e^{-S}$ gives the momentum-space rules

At one loop, the four-point one-particle-irreducible correlation function receives the three bubble diagrams in the $s$, $t$, and $u$ channels. If $P$ is the momentum through one channel, its contribution isUsing a Feynman parameter, shifting the loop momentum, and writing ${\Delta }_x=m^2+x(1-x)P^2$ gives, up to terms caused by shifting the boundary of a hard cutoff,Thus every channel has the logarithmic ultraviolet divergenceThe complete one-loop vertex isLet $I_{m,\mathrm{os}}$ denote the bracket evaluated at the chosen on-shell kinematic point. The on-shell renormalization scheme requires $V_a^{(4)}{|}_{\mathrm{os}}=-{\lambda }_{\mathrm{phys}}$, hence the perturbative solution isThe cutoff dependence of ${\lambda }_a$ is the coupling counterterm needed to hold the measured coupling fixed.

Write ${\varphi }_B=Z^{1/2}\varphi$, $Z=1+{\delta }_Z$, and define the mass counterterm ${\delta }_m$ as a squared-mass coefficient:With the self-energy convention in b, the inverse scalar propagator is $p^2-m^2+{\delta }_Z{p}^2-{\delta }_m-\Sigma (p^2)$. Requiring the pole mass to equal $m$ therefore gives the on-shell renormalization scheme conditionAt one loop order, ${\Sigma }_1^{\prime }=0$; hence a unit pole residue also gives ${\delta }_Z^{(1)}=0$, andThe mass counterterm Feynman rule is $-i{\delta }_m$, cancelling the tadpole diagram. A convention defining a positive mass subtraction by $m_B^2=m^2-{\delta }_m^{\mathrm{sub}}$ instead has ${\delta }_m^{\mathrm{sub}}=-{\delta }_m$; specifying this convention avoids a spurious sign disagreement.

There is no one-loop wave-function renormalization because the tadpole diagram is independent of momentum. Each one-loop four-point bubble diagram has two scalar propagators, so its large Euclidean loop momentum integral behaves as $\int d^2{k}_E/k_E^4$ and is ultraviolet finite. Thus no divergent quartic interaction counterterm is required. Finite coupling redefinitions remain possible. At higher loop orders, finite wave-function renormalization may be needed if unit pole residue is imposed; power counting only rules out its ultraviolet divergence.

Let $I,L,V$ count internal lines, independent loops, and interaction vertices of a connected Feynman diagram. The graph identity $L=I-V+1$ and the two inverse powers from each scalar propagator give its superficial degree of divergence:The same formula applies to every connected subgraph. Graphs with $V\ge 2$ have negative superficial degree of divergence, but this alone does not remove a possible ultraviolet subdivergence. A one-vertex subgraph with external legs in the quartic scalar field theory can diverge only by a tadpole diagram, whose subtraction was fixed in c. Subtract every such tadpole diagram; every remaining loop subgraph then has $V\ge 2$ and negative superficial degree of divergence. Its ultraviolet divergence is absent. Equivalently, normal ordering the quartic interaction removes all self-contractions at a single interaction vertex. Thus no new divergent field-dependent counterterm structure is needed, and in a normal-ordering subtraction scheme in two dimensions the one-loop mass subtraction exhausts these divergent coefficients.

The literal assertion that there are no further counterterm contributions in the on-shell renormalization scheme is stronger and generally false. A concrete counterexample is the two-loop sunset diagram. After analytic continuation below its three-particle threshold, the magnitude of its self-energy at $p^2=m^2$ contains the strictly positive finite integralThe integrable corners do not spoil convergence. Its nonzero value shifts the pole mass, requiring an additional finite mass counterterm if $m$ is to remain the exact pole mass; its derivative can likewise change the pole residue. These are compatible with the power counting result. If vacuum diagrams are retained rather than divided out in normalized correlation functions, a separate constant vacuum energy counterterm is also required: the one-vertex double-loop vacuum diagram is already divergent. The claim about the mass subtraction concerns field-dependent counterterms and normalized correlation functions.

The additional six-leg interaction vertex has Feynman rule $-i{g}_6$. Since both interactions lack derivatives, the graph identity still gives $\omega =2(1-V)$, with $V=V_4+V_6$. The difference is that a one-vertex contraction of the phi-six theory can leave four external legs: choosing the contracted pair gives $\left(\genfrac{}{}{0ex}{}{6}{2}\right)=15$. Its quartic interaction correction isConsequently a one-loop quartic interaction counterterm is now necessary:The one-loop mass counterterm from c is unchanged. Contracting four legs of a single six-leg interaction vertex gives $\left(\genfrac{}{}{0ex}{}{6}{2}\right)3=45$ double contractions, hence a two-loop raw self-energy $g_6{T}_{\Lambda }^2/8$. Its subtraction must include the quartic interaction counterterm insertion, rather than subtracting this raw expression in isolation.

All divergent field-dependent coefficients are most simply organized by the normal-ordering subtraction scheme in two dimensions. With $T=T_{\Lambda }$, express a potential normal ordered at mass $m$ in ordinary fields:Thus its divergent subtraction coefficients, in this scheme, areIndeed the two-loop self-energy terms are $g_6{T}^2/8+\delta g_{4}T/2=-g_6{T}^2/8$, which the displayed mass counterterm cancels. The constant terms only affect vacuum energy. After removing all one-vertex self-contractions, every remaining subgraph is ultraviolet finite. Additional finite on-shell renormalization scheme adjustments are subject to the same qualification as d.

The coefficient of a simple one-particle pole in a propagator is its pole residue. If $G(p)=i/[p^2-m_0^2-\Sigma (p^2)]$ and $m$ is the pole mass, then $Z_{\mathrm{pole}}=[1-{\Sigma }^{\prime }(m^2){]}^{-1}$. An on-shell renormalization scheme may impose unit residue by a finite or divergent wave-function renormalization; mere ultraviolet finiteness does not force the residue to be one.

In a massive two-dimensional quartic scalar field theory, the two-loop sunset diagram is ultraviolet finite but changes the pole mass. Feynman parameters give, below the three-particle threshold,At $s=m^2$ the denominator is positive in the interior of the simplex, and its corners are integrable. Thus the result is finite and nonzero for $g_4\ne 0$. Absence of new divergent counterterms does not imply absence of finite mass counterterms in an on-shell renormalization scheme.