FeynCalc manual (development version)

PaVeLimitTo4

PaVeLimitTo4[expr] simplifies products of Passarino-Veltman functions and $D$-dependent prefactors by evaluating the prefactors at $D=4$ and adding an extra term from the product of $(D-4)$ and the UV pole of the Passarino-Veltman function.

This is possible because the UV poles of arbitrary Passarino-Veltman functions can be determined via PaVeUVPart. The result is valid up to 0th order in Epsilon, i.e. it is sufficient for 1-loop calculations.

Warning! This simplification always ignores possible IR poles of Passarino-Veltman functions. Therefore, it can be used only if all IR poles are regulated without using dimensional regularization (e.g. by assigning gluons or photons a fake mass) or if it is known in advance that the given expression is free of IR singularities.

The application of PaVeLimitTo4 is equivalent to using the old OneLoop routine with the flags $LimitTo4 and $LimitTo4IRUnsafe set to True.

See also

Overview, $LimitTo4.

Examples

ex = (D - 2)/(D - 3) A0[m^2] 
 
PaVeLimitTo4[ex]

$$\frac{(D-2) \;\text{A}_0\left(m^2\right)}{D-3}$$

$$2 \;\text{A}_0\left(m^2\right)+2 m^2$$

Simplify the 1-loop amplitude for $H \to g g$

ex = (-(1/((-2 + D) mH^2 mW sinW)) 2 I (-4 + D) e gs^2 mt^2 \[Pi]^2 B0[mH^2, mt^2, mt^2] 
     SD[Glu2, Glu3] (-2 SPD[k1, Polarization[k2, -I, Transversality -> True]] 
        SPD[k2, Polarization[k1, -I, Transversality -> True]] + 
       mH^2 SPD[Polarization[k1, -I, Transversality -> True], 
         Polarization[k2, -I, Transversality -> True]]) - 1/((-2 + 
          D) mH^2 mW sinW) I e gs^2 mt^2 (-2 mH^2 + D mH^2 - 
       8 mt^2) \[Pi]^2 C0[0, 0, mH^2, mt^2, mt^2, mt^2] SD[Glu2, Glu3] (-2 SPD[k1, 
         Polarization[k2, -I, Transversality -> True]] SPD[k2, Polarization[k1, 
          -I, Transversality -> True]] + mH^2 SPD[Polarization[k1, -I, 
          Transversality -> True], Polarization[k2, -I, Transversality -> True]]))

$$-\frac{2 i \pi ^2 (D-4) e \;\text{gs}^2 \;\text{mt}^2 \delta ^{\text{Glu2}\;\text{Glu3}} \;\text{B}_0\left(\text{mH}^2,\text{mt}^2,\text{mt}^2\right) \left(\text{mH}^2 \left(\varepsilon ^*(\text{k1})\cdot \varepsilon ^*(\text{k2})\right)-2 \left(\text{k1}\cdot \varepsilon ^*(\text{k2})\right) \left(\text{k2}\cdot \varepsilon ^*(\text{k1})\right)\right)}{(D-2) \;\text{mH}^2 \;\text{mW} \;\text{sinW}}-\frac{i \pi ^2 e \;\text{gs}^2 \;\text{mt}^2 \left(D \;\text{mH}^2-2 \;\text{mH}^2-8 \;\text{mt}^2\right) \delta ^{\text{Glu2}\;\text{Glu3}} \;\text{C}_0\left(0,0,\text{mH}^2,\text{mt}^2,\text{mt}^2,\text{mt}^2\right) \left(\text{mH}^2 \left(\varepsilon ^*(\text{k1})\cdot \varepsilon ^*(\text{k2})\right)-2 \left(\text{k1}\cdot \varepsilon ^*(\text{k2})\right) \left(\text{k2}\cdot \varepsilon ^*(\text{k1})\right)\right)}{(D-2) \;\text{mH}^2 \;\text{mW} \;\text{sinW}}$$

PaVeLimitTo4[ex]

$$\frac{i \pi ^2 e \;\text{gs}^2 \;\text{mt}^2 \left(\text{mH}^2-4 \;\text{mt}^2\right) \delta ^{\text{Glu2}\;\text{Glu3}} \;\text{C}_0\left(0,0,\text{mH}^2,\text{mt}^2,\text{mt}^2,\text{mt}^2\right) \left(2 \left(\overline{\text{k1}}\cdot \bar{\varepsilon }^*(\text{k2})\right) \left(\overline{\text{k2}}\cdot \bar{\varepsilon }^*(\text{k1})\right)-\text{mH}^2 \left(\bar{\varepsilon }^*(\text{k1})\cdot \bar{\varepsilon }^*(\text{k2})\right)\right)}{\text{mH}^2 \;\text{mW} \;\text{sinW}}-\frac{2 i \pi ^2 e \;\text{gs}^2 \;\text{mt}^2 \delta ^{\text{Glu2}\;\text{Glu3}} \left(2 \left(\overline{\text{k1}}\cdot \bar{\varepsilon }^*(\text{k2})\right) \left(\overline{\text{k2}}\cdot \bar{\varepsilon }^*(\text{k1})\right)-\text{mH}^2 \left(\bar{\varepsilon }^*(\text{k1})\cdot \bar{\varepsilon }^*(\text{k2})\right)\right)}{\text{mH}^2 \;\text{mW} \;\text{sinW}}$$