Dimensional Regularization as an Amplitude Tool

Dimensional regularization defines a loop integral in a nonempty convergence domain when one exists and continues its parameter or gamma-function representation meromorphically in the spacetime dimension. Scaleless integrals, which have no dimension where both endpoints converge, are fixed separately by analytic continuation and homogeneity. Near four dimensions, ordinary ultraviolet and soft or collinear infrared divergences become Laurent poles in $\epsilon$ while Lorentz covariance and loop-momentum translation invariance remain manifest. The regulator exposes singular behavior; it does not by itself say whether a pole is UV or IR, nor does it renormalize the amplitude.

Required background. Feynman and Schwinger Parameters supplies the parameter and gamma-function representations on which the analytic continuation is based.

The continuation and its normalization

On this page set

$$d=4-2\epsilon$$

and insert $\mu^{2\epsilon}$ per loop so a logarithmically dimensionless four-dimensional integral retains its intended mass dimension. Some conventions additionally attach $e^{\gamma_E\epsilon}(4\pi)^{-\epsilon}$ to each loop. That choice is legitimate, but it must accompany every quoted Laurent coefficient.

After Feynman parameterization and a Wick rotation fixed by the $i0$ prescription, the radial integral is expressed through gamma functions. Analytic continuation then yields the standard Minkowski master formula

$$\mu^{2\epsilon}\int\frac{\mathrm d^d\ell}{(2\pi)^d} \frac{1}{(\ell^2-\Delta+i0)^n} =\frac{i(-1)^n}{(4\pi)^{d/2}} \frac{\Gamma(n-d/2)}{\Gamma(n)} \mu^{2\epsilon}(\Delta-i0)^{d/2-n}.$$

The mass dimension is $d-2n+2\epsilon=4-2n$, as intended. The factor $(-1)^n$ and the phase of $\Delta-i0$ are linked: changing metric or propagator conventions requires changing them together. Dimensional continuation and its convergence domains are developed in Weinzierl 2022, §2.4.2, pp. 28–39.

How poles arise

For $n=2$, the gamma function is $\Gamma(\epsilon)$, and

$$\Gamma(\epsilon)\left(\frac{\mu^2}{\Delta-i0}\right)^\epsilon =\frac1\epsilon-\gamma_E+\log\frac{\mu^2}{\Delta-i0}+O(\epsilon).$$

The $1/\epsilon$ pole is the analytic image of a logarithmic endpoint divergence. Which endpoint matters must be diagnosed before or alongside the continuation:

• large loop momentum or a vanishing Schwinger scale indicates UV behavior;
• soft, collinear, or parameter-boundary configurations can indicate IR behavior;
• a threshold zero of $\Delta$ controls a branch point and imaginary part, not necessarily a Laurent pole.

The same $\epsilon$ regulates both UV and IR singularities. Consequently, the symbol $1/\epsilon$ alone carries no origin label. Off-shell continuation, small masses, region analysis, or subtraction operators are ways to separate the mechanisms.

For the massive bubble of the parameter page,

$$B(p^2)=\frac{i}{(4\pi)^{2-\epsilon}} \Gamma(\epsilon) \int_0^1\mathrm dx \left(\frac{\mu^2}{\Delta(x)-i0}\right)^\epsilon .$$

With the loop prefactor used here its Laurent expansion begins

$$B(p^2)=\frac{i}{(4\pi)^2} \left[ \frac1\epsilon-\gamma_E+\log4\pi +\int_0^1\mathrm dx\, \log\frac{\mu^2}{\Delta(x)-i0} \right]+O(\epsilon).$$

The pole coefficient is independent of $p^2$ and the masses, as required for the logarithmic UV endpoint. The threshold and mass dependence live in the finite parameter integral. Attaching $e^{\gamma_E\epsilon}(4\pi)^{-\epsilon}$ to the loop measure would remove $-\gamma_E+\log4\pi$; conventions cannot be switched after the Laurent expansion.

Why scaleless integrals vanish

Consider a homogeneous integral with no mass or external invariant, such as

$$S=\mu^{2\epsilon}\int\frac{\mathrm d^d\ell}{(2\pi)^d}\frac{1}{(\ell^2+i0)^2}.$$

Under $\ell\mapsto\lambda\ell$, it would obey $S=\lambda^{-2\epsilon}S$ for every positive $\lambda$. Analytic continuation therefore sets $S=0$. This does not mean that its integrand lacks singular regions: in four dimensions it has both an ultraviolet endpoint and an infrared endpoint. Introducing an auxiliary scale can separate equal-and-opposite UV and IR pole contributions whose dimensionally regulated sum is zero.

The vanishing of scaleless integrals and the distinction between the Feynman prescription and the dimensional regulator are stated explicitly in Abreu, Britto, and Duhr 2022, §1.1, pp. 4–5.

A massless bubble check

For spacelike $p^2<0$,

$$B_0(p^2)=\mu^{2\epsilon}\int\frac{\mathrm d^d\ell}{(2\pi)^d} \frac{1}{(\ell^2+i0)((\ell+p)^2+i0)}$$

becomes

$$B_0(p^2)=\frac{i}{(4\pi)^{2-\epsilon}} \Gamma(\epsilon) \int_0^1\mathrm d x \left(\frac{\mu^2}{-x(1-x)p^2-i0}\right)^\epsilon .$$

The parameter integral is $B(1-\epsilon,1-\epsilon)$. Expanding it shows that the coefficient of the UV pole is independent of $p^2$, while the finite part contains $\log(\mu^2/(-p^2-i0))$. If one instead sets $p=0$ before integration, the integral is scaleless and vanishes: that limit merges the UV contribution with a new IR singularity. Limits and analytic continuation therefore need not commute term by term.

Boundaries of the method

Dimensional regularization does not prescribe counterterms or a subtraction scheme. It also need not regulate rapidity divergences between modes of equal virtuality; those require an additional rapidity regulator and subtraction convention. Intrinsically four-dimensional objects such as $\gamma_5$ and Levi-Civita tensors need separate definitions, and different spin-state continuation schemes can differ in intermediate finite terms. These choices must be carried with an amplitude, not silently inferred from $d=4-2\epsilon$.

The tensor algebra is part of the regulator: $\eta^\mu{}_{\mu}=d$ and rotational averaging gives $\ell^\mu\ell^\nu\mapsto\eta^{\mu\nu}\ell^2/d$, not the four-dimensional replacements. A numerator proportional to $d-4=-2\epsilon$ can multiply a $1/\epsilon$ integral and leave a finite rational term. Setting it to zero before integration changes the regulated amplitude.

Exercises

1. Put $n=1$ in the master formula. Its dimension is two and its gamma function is $\Gamma(-1+\epsilon)$, matching the quadratic power counting of a massive tadpole.
2. Explain why $p\to0$ in the massless bubble is not a safe substitution after isolating only the UV pole. The limit creates an IR scale endpoint, and the full scaleless integral is defined as the cancellation of the two analytically continued regions.

Where dimensional continuation leads

• UV/IR Poles and the Renormalized-Amplitude Interface classifies the poles of a regulated amplitude and preserves the data needed downstream.
• Dimensional Regularization and Minimal Subtraction changes the role of dimensional continuation from regulator to a specified ultraviolet subtraction convention.

References

• Abreu, Samuel, Ruth Britto, and Claude Duhr. “The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical Structures in Feynman Integrals.” Journal of Physics A: Mathematical and Theoretical 55 (2022): 443004. doi:10.1088/1751-8121/ac87de.
• Weinzierl, Stefan. Feynman Integrals. Cham: Springer, 2022. doi:10.1007/978-3-030-99558-4. Open PDF.