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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.