with some form of regularization for the soft and collinear singularities. Assuming that we are using dimensional regularization, the phase space $\Phi_B$ and $\Phi_R$ are evaluated in $d = 4 - 2 \epsilon$ dimensions. The value of an observable $\mathcal{O}$ is then given by $$\langle \mathcal{O} \rangle = \int \mathcal{O} {\mathrm{d}}\sigma = \int {\mathrm{d}}\Phi_B (B (\Phi_B) + V (\Phi_B))  \mathcal{O} (\Phi_B) + \int {\mathrm{d}}\Phi_R
  R (\Phi_R) \mathcal{O} (\Phi_R) . \label{eq:observableO}$$ We can think of our observable $\mathcal{O}$ as the cross section in a given histogram bin of some kinematic distribution. Again, in eq. (\[eq:observableO\]) we assume that we have a $d$-dimensional definition for our observable, with the appropriate 4-dimensional limit. If the observable is IR-safe, soft and collinear singularities will cancel in eq. (\[eq:observableO\]), yielding a finite result.

Subtraction method
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In the subtraction method, one introduces a parametrization of the real phase space of the form $\Phi_R = \Phi_R (\Phi_B, \Phi_{{\ensuremath{\operatorname{rad}}}})$, with $${\mathrm{d}}\Phi_R = {\mathrm{d}}\Phi_B {\mathrm{d}}\Phi_{{\ensuremath{\operatorname{rad}}}},$$ where $\Phi_{{\ensuremath{\operatorname{rad}}}}$ has $d - 1$ dimensions, and parametrizes the emission of the extra parton. The parametrization must have a smooth behaviour in the soft and collinear limit. Thus, in the limit of soft emission, the kinematics of all but the soft parton 