- M_W^2)^2 + \Gamma_W^2 M_W^2}\,,\\
  P^4_r & = & \frac{M_Z^4}{(s_{35} - M_Z^2)^2 + \Gamma_Z^2 M_Z^2} \times
  \frac{M_Z^4}{(s_{46} - M_Z^2)^2 + \Gamma_Z^2 M_Z^2} \,,\\
  P^5_r & = & \frac{M_Z^4}{(s_{357} - M_Z^2)^2 + \Gamma_Z^2 M_Z^2} \times
  \frac{M_Z^4}{(s_{46} - M_Z^2)^2 + \Gamma_Z^2 M_Z^2}\,,\\
  P^6_r & = & \frac{M_Z^4}{(s_{35} - M_Z^2)^2 + \Gamma_Z^2 M_Z^2} \times
  \frac{M_Z^4}{(s_{467}^{} - M_Z^2)^2 + \Gamma_Z^2 M_Z^2}\,.\end{aligned}$$ The singular regions $\alpha_r$ are displayed in tab. \[tab:alr-regions-example\].

   $\alpha_r$   $f_r$   emitter   $d^{- 1} (\alpha_r)$
  ------------ ------- --------- ----------------------
       1          1        0          $d^{- 1}_7$
       2          2        3        $d^{- 1}_{37,2}$
       3          2        4        $d^{- 1}_{47,2}$
       4          3        5        $d^{- 1}_{57,3}$
       5          3        6        $d^{- 1}_{67,3}$
       6          4        0          $d^{- 1}_7$
       7          5        3        $d^{- 1}_{37,5}$
       8          5        4        $d^{- 1}_{57,5}$
       9          6        5        $d^{- 1}_{47,6}$
       10         6        6        $d^{- 1}_{67,6}$

  : \[tab:alr-regions-example\]

Notice that the final state radiation $d$ factors carry in the subscript the position of the two partons that become collinear, and, after a comma, an index specifying the resonance history. We are in fact assuming that the $d$ factors are computed in the frame of the resonance that owns the two collinear partons. Notice also that the standard (non resonance aware) `POWHEG` implementation would have found 5 regions, one for the 