Tensor integral reduction

For a Euclidean integral depending only on loop momentum squared, a vector moment vanishes because reversing the integration variable changes its sign. A rank-two moment must be proportional to the Euclidean metric. Contracting its indices fixes the coefficient to one divided by d. This argument assumes a rotationally symmetric regulator or the dimensionally regulated continuation of such an integral. A nonsymmetric cutoff region can produce extra structures and invalidate the shortcut.

The analogous Minkowski statement uses the metric with the declared signature, but Wick-rotation factors and signs must be carried through the full expression. Deriving the identity in a Euclidean domain and continuing consistently is often less error prone. Odd moments vanish only after the denominator has the required symmetry: before shifting a two-denominator bubble, a linear numerator can have a nonzero term proportional to external momentum. Dropping it by an unjustified oddness claim removes a real form factor.

For an equal-mass Euclidean bubble with denominators centered at k and k plus p, transform the integration variable to minus k minus p. The denominators exchange, while the numerator vector becomes minus k minus p. Equating the transformed and original integrals gives the vector coefficient minus one half times the scalar bubble. This derivation requires equal masses and a regulator permitting the shift. Different masses do not obey the same simple exchange relation.

For a rank-two bubble, the available symmetric structures are a metric tensor and two copies of the external momentum. Contracting with the metric and with two momenta gives linear equations for their scalar coefficients. At more external legs, solving analogous systems introduces inverse Gram matrices of external momentum dot products. Near linearly dependent kinematics, those inverses can become numerically large even if the original tensor integral is well behaved. Use specialized limits or alternative reductions instead of trusting subtractive cancellation at inadequate precision.

The factor one over d is one over four plus epsilon over eight plus higher terms. Multiplying it by a scalar moment with a simple pole produces a finite contribution from the epsilon-over-eight term. Setting d equal to four too early misses that contribution. Similar effects occur when Dirac traces or polarization counts depend on dimension. Which dimension is used for internal algebra and external states is part of a regularization scheme and must be stated in gauge-theory calculations.

Reduction also produces scalar numerators that can cancel propagators. For example, loop momentum squared can be written as the denominator minus its mass term in a Euclidean radial integral. The resulting lower-power integrals must retain their separate regularization before recombination. A cancellation of poles is a useful check, but ultraviolet and infrared origins still matter. The vector lab below illustrates a finite-dimensional isotropic average only; it cannot continue d, evaluate a loop, or diagnose a physical Gram determinant.