Loop integral

A loop integral supplies the momentum-space contribution of a closed internal path in a perturbative Feynman diagram. For a one-loop graph, a typical expression has the form ∫d^dk multiplied by propagator denominators such as [(k+p)²−m²+iε]⁻¹. The loop momentum k is not fixed by external momentum conservation and must be integrated over all values. Numerators containing powers of k arise from spin, gauge, or derivative couplings; these are tensor loop integrals, while integrals with scalar numerators are scalar integrals.

Loop integrals first appear at one loop and proliferate rapidly at higher orders. They correct measurable quantities such as anomalous magnetic moments, decay rates, and cross sections, while their divergences are handled within renormalized quantum field theory.²

Regularization makes a divergent loop integral mathematically controllable without changing the desired physical limit. Dimensional regularization replaces four-dimensional integration with integration in d dimensions, conventionally d=4−2ε; ultraviolet divergences then appear as poles in ε. The method preserves Lorentz symmetry and, in many gauge theories, makes gauge-invariant relations easier to maintain.³

Ultraviolet poles describe sensitivity to large virtual momenta and are removed or reorganized by counterterms and renormalization conditions. Infrared divergences instead arise from massless or soft and collinear particles. They may cancel between virtual loop contributions and real-emission contributions for sufficiently inclusive observables, as expressed by the Kinoshita–Lee–Nauenberg framework. A mass, an infrared regulator, or dimensional regularization can expose these singularities, but the regulator must disappear from a properly defined physical prediction.

Modern calculations reduce families of loop integrals to a finite basis of master integrals. Integration-by-parts identities follow from integrating total derivatives in loop momentum, producing linear relations among integrals with different propagator powers and numerators. Lorentz-invariance identities and symmetry relations provide additional reductions. The Passarino–Veltman method performs a particularly useful one-loop tensor reduction by expressing tensor coefficients through scalar functions associated with external momenta and metric tensors.

Master integrals can then be evaluated by Feynman parameters, Mellin–Barnes representations, differential equations, or generalized unitarity. Choosing a basis with uniform transcendental properties can simplify differential equations, turning them into canonical systems whose solutions are iterated integrals or, in more complicated cases, elliptic functions.⁴

Loop integrals are also analytic functions whose singularities encode physical thresholds and possible internal on-shell configurations. The iε prescription specifies how propagator poles are bypassed and selects the causal, or Feynman, boundary value; changing kinematic regions can therefore introduce imaginary parts associated with allowed production channels.

Not every integral belongs to the same function class. One-loop massless examples often reduce to logarithms and polylogarithms, whereas massive multi-loop sunrise-type integrals can require elliptic generalizations. Integration-by-parts reduction is powerful but may become computationally large because sectors with distinct propagator sets must be organized systematically. In gauge theories, individual loop integrals can depend on gauge choice even though the properly assembled observable does not. Cuts of loop integrals provide another perspective: putting selected internal lines on shell relates discontinuities of amplitudes to products of lower-order amplitudes, connecting direct integration with unitarity methods.

Authors commonly label a loop family by denominators D_i and write an integral as I(a₁,…,a_n) = ∫d^dk ∏_iD_i^−a_i. Positive indices denote ordinary propagator powers; zero indices remove denominators, and negative indices encode irreducible scalar products in the numerator. Normalization factors such as iπ^d/2, powers of the renormalization scale, and factors of (2π)^d vary among subfields.

Results must therefore be compared only after conventions, metric signature, masses, external-momentum routing, and branch choices have been aligned. Scalar one-loop functions are often denoted by names such as A, B, C, and D for one-, two-, three-, and four-point families, following standard reduction conventions.

Feynman diagram

A graphical representation of terms in a perturbative expansion of a quantum field theory.

Ultraviolet divergence

A divergence caused by integration over arbitrarily large internal momenta.

Infrared divergence

A divergence associated with low-energy, soft, or collinear massless configurations.

Master integral

An irreducible member of an integral family to which other integrals are reduced by identities.

Dimensional regularization

A regularization method that analytically continues momentum integrals from four dimensions to d dimensions.

Passarino–Veltman reduction

A procedure that expresses one-loop tensor integrals in terms of scalar coefficient functions.