Slicing of the phase space and
treatment of infrared singularities
For a generic partonic reaction 1 + 2 > 3 + 4 + 5, two particles of the
final state, say 3 and 4, have a high p_{T} and are well separated
in phase space, while the last one, say 5, can be soft,
and/or collinear to either of the four others. The phase space is sliced into
four parts using two arbitrary, unphysical parameters p_{Tm} and
R_{th} in the following way:
 Part I:

The norm p_{T5} of the transverse momentum of the particle 5 is
required to be less than p_{Tm}, taken to be small compared to the
other transverse momenta.
The integration over particle 5 in this cylinder supplies the infrared
and initial state collinear singularities.
It also yields a small fraction of the final state collinear singularities.
 Part II a:

The transverse momentum vector of the particle 5 is required to have a norm
larger than p_{Tm}, and to belong to the cone C_{3} about the
direction of particle 3, defined by
(y_{5}  y_{3})^{2} +
(phi_{5}  phi_{3})^{2}
< R_{th}^{2}.
The integration in the cone C_{3} supplies most of the final state
collinear singularities appearing when 5 is collinear to 3, i.e.
those which are not already contained in part I.
 Part II b:

The transverse momentum vector of the particle 5 is required to have a norm
larger than p_{Tm}, and to belong to the cone C_{4} about the
direction of particle 4, defined by
(y_{5}  y_{4})^{2} +
(phi_{5}  phi_{4})^{2}
< R_{th}^{2}.
The integration in the cone C_{4} supplies most of the final state
collinear singularities appearing when 5 is collinear to 4.
(Particle 5 can never be simultaneously in Parts II a and II b)
 Part II c:

The transverse momentum vector of the particle 5 is required to have a
norm larger than p_{Tm}, and to belong to neither of the two cones
C_{3}, C_{4}. This slice yields no divergence, and can thus be
treated directly in 4 dimensions.
The user has to specify the actual values of p_{Tm} and R_{th}
she or he wishes to use.
The soft and collinear singularities appearing on Parts I, II a and II b are
first regularized by dimensional continuation of the spacetime dimension from
4 to d = 4  2 epsilon with epsilon < 0. Then, the ddimensional
integration over the kinematic variables (transverse momentum, rapidity and
azimuthal angles) of the particle 5 is performed analytically.
This materializes the regularized infrared singularities as poles in 1/epsilon,
which come together with, in particular, p_{Tm} and/or R_{th}
dependent terms.
Over Parts II a and II b, the phase space slicing method is supplemented by
a subtraction method in order to keep the exact, full dependence on
R_{th} (this slight modification is relevant to improve the numerical
precision of the calculations). On the other hand the p_{Tm} dependence
is kept everywhere up to terms O(p_{Tm} ln(p_{Tm})) only.
The fate of the p_{Tm}, R_{th} dependent terms is discussed in
Cancellation of the p_{Tm} and R_{th} dependences in physical cross sections.
After combination of these contributions of real emission with the virtual
corrections, the soft singularities cancel, and the remaining collinear
singularities which do not cancel are factorized and absorbed in parton
distribution or fragmentation functions. This is done once and for all in the
code, the user no longer has to worry about this.
The resulting quantities are finite. They correspond to "pseudo" cross
sections where the hard partons in the parts I, II a and II b are unresolved
from the soft or collinear parton 5 which has been "integrated out"
inclusively on these parts. The word "pseudo" means that they are not genuine
cross sections, as they are not positive in general. They also separately
depend on p_{Tm} and/or R_{th}. They are split for computational
convenience into two kinds:

We call `pseudo cross section for some 2 > 2 process' the sum of
the lowest order term plus the fraction of the corresponding virtual
corrections where the infrared and collinear singularities have been
subtracted, and which have the same kinematics as a genuine 2 > 2
process.

The contributions where the uncanceled collinear singularies are
absorbed into parton distribution functions (on part I) or fragmentation
functions (on parts II a and II b) involve an extra convolution over a
variable of collinear splitting, as compared to the kinematics of a
genuine 2 > 2 process.
We call them `pseudo cross sections for quasi 2 > 2 processes'.
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