Regularization in quantum field theory

Regularization modifies divergent loop integrals by introducing an auxiliary parameter. The regulator is removed after its dependence has been absorbed into counterterms.

An ultraviolet divergence comes from arbitrarily large loop momentum, equivalently from singular short-distance behavior. Locality makes its regulator dependence cancellable by local counterterms in a renormalizable theory.

An amplitude is ultraviolet finite if its regulated high-loop momentum contribution has a finite regulator-removal limit after whatever lower-order subtractions are specified. A zero superficial degree of divergence does not force divergence: gauge invariance can cancel the candidate local term. Conversely, superficial convergence alone does not exclude divergent subgraphs. Ultraviolet finiteness makes no claim about infrared divergences or physical threshold singularities.

An ultraviolet divergence appears as a pole of a dimensionally continued Feynman integral. In $d=4-2ϵ$, a logarithmically divergent one-loop integral typically has a $1/ϵ$ term. Dimensional regularization can also produce infrared divergences with the same formal regulator, so the origin must be identified before interpreting a pole as an ultraviolet counterterm.

An ultraviolet subdivergence arises when loop momenta internal to a proper subgraph tend to infinity while the momenta entering that subgraph are held fixed. It can occur even if simultaneous scaling of all momenta of the full Feynman diagram is convergent. Recursive local counterterms subtract these subgraph divergences before the remaining overall divergence is analyzed. A tadpole diagram inserted into an otherwise convergent six-point phi-fourth theory Feynman diagram is a simple massive example.

Cutoff regularization restricts loop momenta to $|\ell |<\Lambda$. The ultraviolet cutoff $\Lambda$ makes individual integrals finite while displaying power and logarithmic ultraviolet divergences explicitly.

Dimensional regularization analytically continues loop integrals from an integer spacetime dimension to $d=d_0-ϵ$. Ultraviolet logarithms then appear as poles in $1/ϵ$.

With an invariant integration measure and no external vector in $f(q^2)$, the rank-two integral must be proportional to the metric. Contracting its indices fixes the coefficient to $1/d$. In dimensional regularization this dimension is the continued $d$, rather than four; retaining its dependence matters when multiplying a divergent loop integral.

Include the measure factor $(2\pi {)}^{-D}$ and a conventional renormalization scale factor in the displayed integral. The Lorentz invariant replacement $k_{\mu }{k}_{\nu }\mapsto g_{\mu \nu }{k}^2/D$ must be performed before taking $D\to 4$. For $D=4-\epsilon$, its scalar integral has pole $i/(8{\pi }^2\epsilon )$, while $4/D-1=\epsilon /4+O({\epsilon }^2)$. Their product has the finite value shown. Setting $D=4$ before integration discards this value and can spoil a Ward identity, even in a final ultraviolet-finite scattering amplitude. This is an elementary example of a vanishing regulator factor multiplying a divergent integral.

For an overall logarithmically divergent one-loop Feynman integral, its leading large-momentum term fixes its ultraviolet pole. Giving its denominators a positive auxiliary mass $M$ regulates the artificial small-momentum singularity produced by setting external scales to zero. The difference from the original leading expansion must be ultraviolet convergent. This justifies extracting its ultraviolet pole without mistaking a scaleless integral for an ultraviolet-finite one; multiloop subdivergences require additional treatment.

A Feynman integral with no nonzero mass or external momentum scale is assigned zero by dimensional regularization. This does not imply separate absence of ultraviolet divergences and infrared divergences. For a logarithmically scaleless example, zero can encode cancellation of the two pole contributions. Keeping a scale or using auxiliary-mass extraction of ultraviolet poles is necessary when only the ultraviolet part is sought.

For $m^2>0$ and $n>d/2$, Schwinger parameterization and a Gaussian integral give $I_{d,n}(m)=\int d^{d}k(2\pi {)}^{-d}(k^2+m^2{)}^{-n}=\Gamma (n-d/2)(m^2{)}^{d/2-n}/[(4\pi {)}^{d/2}\Gamma (n)]$. Its meromorphic continuation supplies dimensional regularization poles.

The one-propagator Euclidean loop momentum integral associated with a tadpole diagram. Dimensional regularization gives $T_d(m)=\Gamma (1-d/2)(m^2{)}^{d/2-1}/(4\pi {)}^{d/2}$. Near $d=3$ this has no pole and $T_3(m)=-m/(4\pi )$ for $m>0$. A momentum-cutoff definition instead contains a linear ultraviolet divergence; the absence of a dimensional-regularization pole here concerns logarithms, not the positivity or convergence of the unregularized integral.

For positive $m^2$, a Feynman parameter and a momentum shift turn the integral into ${\int }_0^1{I}_{d,2}(\sqrt{m^2+x(1-x)p^2})dx$. The Euclidean massive loop integral has factor $\Gamma (ϵ/2)=2/ϵ+O(1)$, while its other factors tend to $(4\pi {)}^{-2}$. The pole is independent of the external momentum and the mass, so it is removable by a local counterterm.

The minimal subtraction scheme chooses counterterms that remove only the poles in the dimensional regulator $ϵ$, without additional finite terms.

In the minimal subtraction scheme, let ${\lambda }_B={\mu }^{ϵ}F$, $F=\lambda +{\sum }_{k\ge 1}{f}_k(\lambda )/{ϵ}^k$, and $m_B^2=m^{2}H$, $H=1+{\sum }_{k\ge 1}{b}_k(\lambda )/{ϵ}^k$. Define the renormalization-group beta function at fixed bare parameters by $\hat{\beta }=\mu d\lambda /d\mu =-ϵ\lambda +\beta (\lambda )$, and the running mass function by ${\gamma }_{m^2}=\mu d\log m^2/d\mu$.

The bare coupling is scale independent, so $0=ϵF+\hat{\beta }{F}^{\prime }$. Its finite coefficient is $f_1+\beta -\lambda f_1^{\prime }$, giving the first formula. Similarly $0={\gamma }_{m^2}H+\hat{\beta }{H}^{\prime }$ has finite coefficient ${\gamma }_{m^2}-\lambda b_1^{\prime }$. The remaining poles cancel recursively: $(\lambda {\partial }_{\lambda }-1)f_{k+1}=\beta f_k^{\prime }$ and $\lambda b_{k+1}^{\prime }={\gamma }_{m^2}{b}_k+\beta b_k^{\prime }$. Thus the simple poles determine the finite renormalization group functions, while higher poles obey consistency conditions.

With ${\beta }_0=-ϵg_0$, the chain rule gives $\beta (g)={\beta }_{0}dg/d{g}_0$. Inverting the displayed coupling expansion yields $g_0=g-a_1{g}^2/ϵ+(2a_1^2-b_1)g^3/{ϵ}^2$. Substitution gives $\beta =-ϵg-a_1{g}^2+2(a_1^2-b_1)g^3/ϵ+O(g^4)$, so finiteness requires the displayed double-pole condition. An omitted simple-pole term $c_1{g}_0^3/ϵ$ supplies a finite $-2c_1{g}^3$ term; higher-pole consistency does not by itself determine the finite two-loop beta function.

The modified minimal subtraction scheme removes the dimensional pole together with the universal constants conventionally grouped as $-\gamma +\log 4\pi$. This simplifies renormalization-group formulas while differing from minimal subtraction only by finite parameter redefinitions.