Cancellation of the pTm and Rth dependences in physical cross sections

Together with the singularities, the integration over the phase space of particle 5 produces in particular the appearence of ln(pTm) and
ln2(pTm) terms in Part I, as well as ln(pTm) ln(Rth) and ln(Rth) terms in Parts II a and II b,

plus terms which vanish as pTm --> 0. These logarithmic terms are extracted analytically.

Furthermore, the subtraction method used in C3 and C4 allows to keep the full Rth dependence, in particular the terms
O(Rth2n) ln(pTm)) to all orders n in Rth in C3 and C4.

On the other hand, terms vanishing with pTm, of order O(pTm ln(pTm)) or smaller, are dropped.

Besides, when the inelastic configurations generated on Part II c are projected onto any inclusive observable, dependences of the corresponding
contribution on pTm and Rth are numerically generated.

The default choice for pTm in the input file is 0.05 pTinf, pTinf being the lowest of the pT of the two hard particles produced.

The parameters pTm and Rth are unphysical, i.e. they need not be matched to any definition of any observable. On the contrary, the result for any observable has to be independent of the actual choice made for the unphysical parameters Rth and pTm when all contributions from 2 --> 2, quasi 2 --> 2, and inelastic (genuine 2 --> 3) are added, although each of them separately does have a strong dependence. The cancellation of the Rth dependence is exact as a matter of principle of the subtraction method; yet it proceeds through numerical compensations. The dependence on pTm of the sum is very small at least when small enough values are used. The user should check these cancellations in her or his calculations, separately for each of the three mechanisms.

Typically for example, for DIPHOX, in the Tevatron and LHC energy ranges, with pTinf of the order of a couple of tens of GeV, when pTm is varied from 0.01 to 0.5 GeV, the NLO contributions to "one-" and "two fragmentation" vary by less than 1% and 5% respectively (the precise calculation of the "two fragmentation" contribution takes a lot of time. In this case the 5% accuracy indicated here actually reflects essentially statistical uncertainties, rather than a slow convergence as pTm is reduced. A better test of stability would be obtained at the expense of a more extensive computation). Similarly, the q qbar and gg initiated processes contributing to the "two direct" mechanism at NLO vary by less than a few percent when considered separately.

An explicit example of these cancellations is given in the appendix B of Eur. Phys. J. C16, 311 (2000) [hep-ph/9911340]

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