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 $D-4 \to 0$.

A generic IR-finite Passarino-Veltman function $X$ can be written as $X = \frac{a}{D-4} + b + \mathcal{O}(\varepsilon)$, with $a$ being the prefactor of the pole and $b$ being the finite part. Therefore, products of such functions with coefficients that are rational functions of $D$ with $f(D) = f(4) + (D-4) f'(4) + \mathcal{O}(\varepsilon^2)$ can be simplified to $f(D) X = f(4) X + a f'(4) + \mathcal{O}(\varepsilon)$, 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 $A$ and $B$ functions. Although $B$ functions can exhibit an IR divergence, such integrals are zero in dimensional regularization, so that no mixing of $\varepsilon$-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

$$\text{False}$$