), -p\_T ( ), ), p\_g = (E\_g, p\_T ( ), p\_T ( ), ), where $E_x$ is the energy of particle $x$, and $p_T$ and $\phi$ are the transverse momentum and azimuthal angle of the gluon.

The Lorentz invariant phase space element, ${\mathrm{d}}\Phi_R$, describing the emission of the additional gluon is obtained from the relation \_3 = \_2 \_R , where \_N = (2 )\^4 \^4 (p\_t - \_[i=1]{}\^N p\_i ) \_[i=1]{}\^[N]{} and $\mathbf{p_i}$ is the three-momentum of particle $i$. We choose to parameterize the radiative phase space in terms of the transverse momentum, $p_T$, rapidity, $y$, and azimuthal angle, $\phi$, of the gluon and so find \_R = J p\_T dy , where the Jacobian factor, $J$, is [^2] J = . This parametrization has the advantage of simplifying the Heaviside function in the POWHEG Sudakov form factor to a lower limit in the integration over $p_T$.

The final components required for the implementation of the POWHEG Sudakov form factor in Eq. \[eq:POSudakov2\] are the dipole functions, $\mathcal{D}_i$, which describe the singular behaviour of the real-emission matrix element. We use the dipole functions defined in the Catani-Seymour subtraction scheme, details of which can be found in [@Catani:1996vz; @Catani:2002hc], 