Calculation of the polarization tensors ofZ→3γandγγ→γγviaW-boson loops in the standard model

The polarization tensors of $Z\to 3\gamma$ and $\gamma \gamma \to \gamma \gamma$ via $W$-boson loops are calculated in the standard model. The constrained equations for the tensors are deduced from crossing symmetry and gauge invariance. These equations are numerically checked by the calculated tensor of $\gamma \gamma \to \gamma \gamma$ to the first three lowest orders in its low-energy expansion.