= Solution
Loop corrections in <perturbative quantum field theory> integrate over unbounded internal <four-momentum>. For example, after a <Wick rotation>, a four-dimensional integral with large-momentum behavior
$$
\int^\Lambda\frac{d^4k_E}{(k_E^2+m^2)^2}
\sim\int^\Lambda\frac{dk_E}{k_E}
$$
has a logarithmic <ultraviolet divergence>. This is sensitivity to short distances, distinct from a possible low-momentum <infrared divergence>.
<Regularization in quantum field theory> first makes the expression mathematically well defined, for example with a cutoff $\Lambda$ or <dimensional regularization> in $d=4-2\epsilon$. A regulator is an intermediate prescription rather than a measured physical parameter; a hard <momentum> cutoff can obscure gauge identities, so their preservation or restoration must be checked. Introduce bare fields and parameters, such as $\phi_0=Z_\phi^{1/2}\phi$, and split the regulated <Lagrangian density> into a renormalized part and local <counterterms>. For instance, <wavefunction>, <mass> and coupling counterterms cancel the corresponding divergent parts of Green functions order by order.
<Renormalization conditions> fix finite parameters through specified observables or a subtraction prescription. Once these physical inputs are held fixed, predictions must not depend on the arbitrary regulator. In a <renormalizable quantum field theory>, a finite set of operator coefficients suffices to absorb ultraviolet divergences at every perturbative order, with the allowed structures constrained by the theory's <symmetries>. Gauge theories impose relations among counterterms through identities such as the <Ward-Takahashi identity>; one cannot choose every diagram's subtraction independently.
The remaining dependence on a chosen <renormalization scale> describes <running couplings> and is organized by the <renormalization group>. An <effective field theory> can instead include higher-dimension operators suppressed by a high scale. Infinitely many are allowed in principle, but only finitely many contribute at a fixed accuracy in the low-energy expansion. Renormalization is therefore a controlled relation between parameters and predictions at different scales, rather than an instruction to discard every infinity without fixing physical inputs.
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