FeynCalc manual (development version)

$LimitTo4

$LimitTo4 is a variable with default setting False. If set to True, the limit Dimension -> 4 is performed after tensor integral decomposition.

$LimitTo4 is a global variable that determines whether UV-divergent Passarino-Veltman functions are simplified by taking the limit D40D-4 \to 0.

A generic IR-finite Passarino-Veltman function XX can be written as X=aD4+b+O(ε)X = \frac{a}{D-4} + b + \mathcal{O}(\varepsilon), with aa being the prefactor of the pole and bb being the finite part. Therefore, products of such functions with coefficients that are rational functions of DD with f(D)=f(4)+(D4)f(4)+O(ε2)f(D) = f(4) + (D-4) f'(4) + \mathcal{O}(\varepsilon^2) can be simplified to f(D)X=f(4)X+af(4)+O(ε)f(D) X = f(4) X + a f'(4) + \mathcal{O}(\varepsilon), whenever such products appear in the reduction.

This relation is correct only if the Passarino-Veltman functions have no IR divergences, or if such divergences are regulated without using dimensional regularization.

For this reason, even when $LimitTo4 is set to True, the simplifications are applied only to AA and BB functions. Although BB functions can exhibit an IR divergence, such integrals are zero in dimensional regularization, so that no mixing of ε\varepsilon-terms from IR and UV can occur.

The default value of $LimitTo4 is False. Notice that even when the switch is set to True, it will essentially affect only the Passarino-Veltman reduction via PaVeReduce.

The modern and more flexible way to simplify amplitudes involving IR-finite PaVe functions is to use the special routine PaVeLimitTo4.

See also

Overview, PaVe, PaVeReduce, OneLoop, $LimitTo4IRUnsafe, PaVeLimitTo4.

Examples

$LimitTo4

False\text{False}