$\mathcal{D}_i $, each of which describes the behaviour of the real-emission matrix element in a singular region of phase space, i.e. when the emitted parton becomes soft or collinear to one of the legs in the Born process. By doing so, the different singular regions can be separated such that Eq. \[eq:POSudakov\] becomes a product of Sudakov form factors \^[PO]{}(p\_T) = \_i ( - d\_R (p\_T (\_R ) -p\_T ) ), \[eq:POSudakov2\] each of which describes the non-emission probability in a particular region of phase space specified by the dipole function $\mathcal{D}_i$.

When applying the POWHEG method to a parton shower ordered in transverse momentum, the hardest emission is generated first using the POWHEG Sudakov form factor in Eq. \[eq:POSudakov2\]. Subsequent emissions are generated with the normal parton shower Sudakov given in Eq. \[eq:PSSudakov\], with the requirement that no parton shower emission has higher transverse momentum than the emission described by $R(\Phi_N, \Phi_R)$. However, to allow QCD coherence effects to be included, an angularly ordered parton shower is used in . Ordering the parton shower in terms of an angular variable means the first emission in the shower may not be the hardest. The POWHEG approach can be reconciled with 