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 pT 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 pTm and Rth in the following way:

Part I:
The norm pT5 of the transverse momentum of the particle 5 is required to be less than pTm, 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 pTm, and to belong to the cone C3 about the direction of particle 3, defined by

(y5 - y3)2 + (phi5 - phi3)2 < Rth2.

The integration in the cone C3 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 pTm, and to belong to the cone C4 about the direction of particle 4, defined by

(y5 - y4)2 + (phi5 - phi4)2 < Rth2.

The integration in the cone C4 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 pTm, and to belong to neither of the two cones C3, C4. This slice yields no divergence, and can thus be treated directly in 4 dimensions.

The user has to specify the actual values of pTm and Rth 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 space-time dimension from 4 to d = 4 - 2 epsilon with epsilon < 0. Then, the d-dimensional 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, pTm and/or Rth 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 Rth (this slight modification is relevant to improve the numerical precision of the calculations). On the other hand the pTm dependence is kept everywhere up to terms O(pTm ln(pTm)) only. The fate of the pTm, Rth dependent terms is discussed in Cancellation of the pTm and Rth 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 pTm and/or Rth. They are split for computational convenience into two kinds:

  1. 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.
  2. 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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