NNLO splitting functions in perturbative QCD
Unless another scheme is explicitly stated, the formulae given below are
in MS.
For the polarized case, which involves the issue of γ5
in dimensional regularization, see Matiounine, Smith and van Neerven,
hep-ph/9803439 = Phys. Rev. D58 (1998) 076002 and our arXiv:1409.5131
(linked below).
The Fortran routines of the exact expressions use the package of
Gehrmann and Remiddi for the harmonic polylogarithms (HPLs)
published in hep-ph/0107173 = CPC 141 (2001) 296.
- 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
( .ps.gz and
.pdf files −
one sign typo in (3.10) corrected )
- The files
xpij2e.f for the exact and
xpij2p.f for the parametrized x-space
flavour-singlet quantities
Pps = Pqq - P+, Pqg ,
Pgq and Pgg as published in
hep-ph/0404111 = Nucl. Phys. B691 (2004) 129-181
( .ps.gz and
.pdf files )
- N-space subroutines for the parametrized non-singlet and singlet
expressions are provided by p2mom.f
- FORM files with the complete Mellin space (even or odd N) and x-space
results for the non-singlet and
singlet cases
Two-loop Fortran routines in HPL notation can be found in
xpns1e.f and xpij1e.f
together with the leading-order functions. The NLO N-space formulae are
available via the QCD-Pegasus package.
Besides above journal papers, the results were 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
( .ps.gz and .pdf files )
Approximations for the 3-loop splitting functions (2000, obsolete)
- The non-singlet and singlet Fortran routines
xpns2n.f and
xpij2n.f --
superseded by the above complete results.
These 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.
These approximations were used in the
2002 evolution benchmarks, so they might still
be useful for checks or debugging.
- The Fortran files
xdPij2M.f for the exact and
xdPij2p.f for the parametrized x-space
singlet quantities
ΔPqq (exact only),
ΔPps = ΔPqq - ΔP+,
ΔPqg , ΔPgq and ΔPgg
as presented in
arXiv:1409.5131 ( .ps.gz and
.pdf files )
- Due to the relation ΔP± = P∓
between the unpolarized and polarized non-singlet splitting functions,
the parametrized ΔPqq is obtained by adding the
corresponding function in xpns2n.f (see above)
to ΔPps . Note: ΔPv has not been
calculated yet.
- An N-space subroutine for the parametrized flavour-singlet quantities
is dp2mom.f; the non-singlet contribution to
ΔPqq is P2MINN in p2mom.f
(this file also provides the logarithmic derivatives of the
Γ-function).
- FORM files with the exact expressions at LO, NLO and NNLO are available
in N-space (for odd N)
and x-space.
The two-loop results in terms of HPLs are given in
xdPij1M.f, and the leading-order quantities in
xdPij0.f. Also here Fortran routines of the LO and
NLO N-space splitting functions are included in the
QCD-Pegasus package.
The quantities ΔPps and ΔPqg were
calculated much early via the photon-exchange structure function
g1, see arXiv:0807.1238.
A conference account of the (quite different) determination of also
ΔPgq and ΔPgg at NNLO can be found in
arXiv:1405.3407.
Photon-parton splitting functions (2005, …)
- 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
( .ps.gz and
.pdf files )
- The subroutine xpgamp.f for the complete results
presented hep-ph/0511112
(Acta Phys. Polon. B37 (2006) 683-687), where Pns,γ and
Pgγ are represented by parametrizations, while the
short exact expression is given for Pps,γ
A full-length paper will all exact expressions is still not available.
However, together the two articles above contain all information
required for carrying out NNLO analyses in both the
MS and DISγ
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 and .pdf files linked above.
The NNLO `time-like' (fragmentation) splitting functions Pt(2)
were determined using analytic-continuation relations to the `space-like'
(parton distribution) case in
hep-ph/0604053 = Phys. Lett. B638 (2006) 61-67
( .ps.gz and .pdf files )
for the non-singlet quantities Pt±,v
in arXiv:0709.3899 = Phys. Lett. B659 (2008) 290-296
( .ps.gz and .pdf files )
for Ptps and Ptgg
and, up to a small remaining uncertainly, in
arXiv:1107.2263 = Nucl. Phys. B854 (2012) 133-152
( .ps.gz and .pdf files )
for Ptqg and Ptgq.
- Fortran routines
xpns2te.f (exact) and
xpns2tp.f (parametrized)
for the non-singlet splitting functions in x-space,
- and xpij2te.f and
xpij2tp.f
for the corresponding flavour-singlet quantities
- N-space subroutines for the parametrized non-singlet and singlet
expressions are provided by pt2mom.f
- A FORM file with the complete x-space and
N-space expressions up to the third order ( except for Ptv
which is identical to Pv ).
xpns1te.f and xpij1te.f
are Fortran routines for the NLO timelike splitting functions in x-space.
A marginally updated ancient NLO N-space routine, prepared for the
1993 GRV fragmentation paper, can be found
here.
The LO expressions are identical to those of the spacelike case for the chosen
normalization -- in which
Dq in Eq. (3) in arXiv:1107.2263 should
have an additional prefactor 1/(2nf).