Loop Integrals and Divergences

At tree level every amplitude is finite: the internal momenta are fixed by the external ones. Loops change this — each closed loop carries an undetermined momentum integrated over all of spacetime, $\int d^4\ell /(2\pi {)}^4$, and these integrals are generically divergent in the ultraviolet (large $\ell$). This page classifies the divergences by power counting and works the canonical one-loop examples, setting up regularization and renormalization.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

Where the loop integral comes from

Each Feynman diagram imposes momentum conservation at every vertex via ${\delta }^4$ functions. For a tree diagram these fix all internal momenta. For a diagram with $L$ independent loops, $L$ internal momenta remain unfixed and must be integrated:

$$L=I-V+1,$$

where $I$ = internal lines and $V$ = vertices. Each loop contributes $\int \frac{d^4\ell }{(2\pi {)}^4}$, and the integrand is a product of propagators $\sim 1/{\ell }^2$ at large $\ell$. Whether the integral converges is a matter of counting powers.

Superficial degree of divergence

For a diagram, define the superficial degree of divergence $D$ as the net power of loop momentum in the numerator minus denominator, counting $d^4\ell$ as $+4$ per loop:

$$\boxed{D=4L-2I_b-I_f+(\text{derivative couplings}),}$$

where $I_b,I_f$ are internal boson/fermion lines (bosons $\sim 1/{\ell }^2$, fermions $\sim 1/\ell$). The integral behaves at large $\Lambda$ as:

"Superficial" because a diagram with $D<0$ can still diverge through a subdiagram with $D\ge 0$; the full statement (Weinberg's theorem) is that a diagram converges iff $D<0$ for the whole diagram and every subdiagram. Nested divergences are handled systematically by BPHZ.

Power counting and renormalizability

Expressing $D$ through the external lines and the coupling's mass dimension gives the central result. In four dimensions, for a theory whose couplings have mass dimension $[g_i]=4-d_i$,

$$D=4-E_b-\frac{3}{2}{E}_f-\sum _{\text{vertices}}(d_i\text{ contributions}),$$

with $E_b,E_f$ the external boson/fermion lines. The decisive fact:

• If every coupling has $[g_i]\ge 0$ (dimension $\le 4$ operators), $D$ depends only on the external legs, so only finitely many amplitudes diverge — the theory is renormalizable.
• If any coupling has $[g_i]<0$ (dimension $>4$ operator), inserting more vertices raises $D$, so infinitely many amplitudes diverge — the theory is non-renormalizable (an effective theory).

This is why the Standard Model is built from operators of dimension $\le 4$: it is the renormalizability criterion in disguise.

Worked example: the ${\varphi }^4$ self-energy

The one-loop correction to the scalar propagator in ${\varphi }^4$ theory is the "tadpole" bubble with one loop, $L=1$, two internal… actually one internal line closing on the quartic vertex:

$$\Sigma \sim \frac{-i\lambda }{2}\int \frac{d^4\ell }{(2\pi {)}^4}\frac{i}{{\ell }^2-m^2+iϵ}.$$

Power counting: $D=4-2=2$, a quadratic divergence $\sim {\Lambda }^2$. It contributes a (divergent) shift to the mass — the origin of mass renormalization and of the hierarchy problem (the scalar mass is not protected by any symmetry). The $\frac{1}{2}$ is the symmetry factor.

Worked example: the vertex correction

The one-loop correction to the four-point vertex has $L=1$ and two internal propagators:

$${\Gamma }_{1\text{-loop}}^{(4)}\sim (-i\lambda {)}^2\int \frac{d^4\ell }{(2\pi {)}^4}\frac{i}{{\ell }^2-m^2}\frac{i}{(\ell +p{)}^2-m^2}.$$

Power counting: $D=4-4=0$, a logarithmic divergence $\sim \ln \Lambda$. This $\ln$ is the seed of the running coupling and the renormalization group: the logarithm's argument is a ratio of scales, and demanding physics not depend on the arbitrary reference scale forces $\lambda$ to run.

Feynman parameters and Wick rotation

Two standard tools tame these integrals before regularizing:

• Feynman parameters combine multiple propagator denominators into one: $$\frac{1}{AB}={\int }_0^{1}dx\frac{1}{[xA+(1-x)B{]}^2},$$ after which the loop momentum can be shifted to complete the square, leaving an integral of the form $\int d^4\ell ({\ell }^2-\Delta {)}^{-n}$.
• Wick rotation ${\ell }^0\to i{\ell }_E^0$ (the same rotation as in the path integral) turns the Minkowski integral into a Euclidean one, $\int d^4{\ell }_E$, with rotational symmetry that makes the radial integral elementary.

The Euclidean integral $\int d^4{\ell }_E({\ell }_E^2+\Delta {)}^{-n}$ then exhibits its divergence explicitly as the upper limit $\to \infty$, ready for regularization.

Summary

• Each loop = one undetermined $\int d^4\ell$; $L=I-V+1$.
• Superficial degree of divergence $D$ counts net powers of $\ell$: $D\ge 0$ diverges (${\Lambda }^D$ or $\ln \Lambda$).
• Couplings of dimension $\ge 0$ ⇒ finitely many divergent amplitudes ⇒ renormalizable.
• Feynman parameters + Wick rotation reduce loops to Euclidean radial integrals.

Where this leads

• Making the divergences finite: regularization.
• Absorbing them into parameters: renormalization and counterterms.
• The logarithms and running: the renormalization group.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 6.3, 10.1.
• Weinberg, The Quantum Theory of Fields, Vol. 1, Ch. 12.1.
• Srednicki, Quantum Field Theory, Ch. 14, 16–18.