NNLO splitting functions in perturbative QCD
Unless another scheme is explicitly stated, the formulae linked below are
given in MS.
- The Fortran files
xpns2e.f for the exact and
xpns2p.f for the parametrized x-space
non-singlet combinations P+ , P- and
Ps = Pv - P- as published in
hep-ph/0403192 (Nucl. Phys. B688 (2004)
101-134)
- The files
xpij2e.f for the exact and
xpij2p.f for the parametrized x-space
singlet quantities
Pps = Pqq - P+, Pqg ,
Pgq and Pgg as published in
hep-ph/0404111 (Nucl. Phys. B691 (2004) 129-181)
- The N-space subroutines for the parametrized non-singlet and singlet
expressions are provided by
p2mom.f
- FORM files with the complete N-space and x-space results for the
non-singlet and singlet
cases
The exact numerical routines require the package of Gehrmann and Remiddi for the
harmonic polylogarithms (HPLs) published in hep-ph/0107173 = CPC 141 (2001) 296.
The one- and two-loop results in the HPL notation can be found in
xpns1e.f and xpij1e.f.
Besides above journal papers, the results are also discussed in the conference
accounts hep-ph/0407321 and
hep-ph/0408075.
Earlier partial results can be found in
hep-ph/0209100 (Nucl. Phys. B646 (2002) 181-200).
Approximations for the 3-loop splitting functions (obsolete)
- The non-singlet and singlet Fortran routines
xpns2n.f and
xpij2n.f --
superseded by the above complete results.
These expression are improved updates, presented and briefly discussed in
hep-ph/0007362 (Phys. Lett. B490 (2000) 111-118),
of the earlier approximations published in
hep-ph/9907472 (Nucl. Phys. B568 (2000) 263-286) and
hep-ph/0006154 (Nucl. Phys. B588 (2000) 345-373),
respectively, for the non-singlet and singlet splitting functions.
- The Fortran file xppf2p.f with the approximate
unpolarized NNLO photon-parton splitting functions -- now superseded,
see below -- published in
hep-ph/0110331 (Nucl. Phys. B621 (2002) 413-458)
- The subroutine xpgamp.f for the complete results
presented hep-ph/0511112
(Acta Phys. Polon. B37 (2006) 683-687), where Pns,&gamma and
Pg&gamma are represented by parametrizations, while the
short exact expression is given for Pps,&gamma
The full-length paper will all exact expressions is not yet available.
However, together the two articles above contain all information
required for carrying out NNLO analyses in both the
MS and DIS&gamma
factorization schemes.
For the first paper on the latter, and the NLO splitting functions, see
Phys. Rev. D45 (1992) 3986-94.
Corresponding results for photon fragmentation are discussed in
Phys. Rev. D48 (1993) 116-128
(a misprint is corrected here).
NOTE: a misprint in Eq. (5.8) of hep-ph/0110331 has been corrected in the
ps-file linked above.
- The Fortran file xptsdiff.f for the differences
of the `time-like' (fragmentation) and `space-like' (parton distribution)
evolution kernels up to NNLO. Only the exact expressions in terms of HPLs
(see above) are available
- A FORM file including these differences
(besides the coefficient functions for e+ e- -->
h + X) as obtained in hep-ph/0604053
(Phys. Lett. B638 (2006) 61-67)
The singlet NNLO splitting functions for the time-like case have not been
calculated yet.