Higgs boson production at the LHC: NNLO partonic cross sections through
order $\epsilon$ and convolutions with splitting functions to N$^3$LO
by
Maik H\"oschele, Jens Hoff, Alexey Pak, Matthias Steinhauser, Takahiro Ueda
arXiv:1211.6559, SFB/CPP-12-93, TTP12-45
Information is provided about the analytical results obtained/used
in the above paper.
sig_tilde_LO_NLO_NNLO.m
=======================
contains analytical results for the partonic cross sections
up to NNLO expressed in terms of HPLs and plus distributions:
rsigc["LO", "gg"] up to O(ep^3)
rsigc["NLO", "gg"] up to O(ep^2)
rsigc["NLO", "qb"] up to O(ep^2)
rsigc["NLO", "qg"] up to O(ep^2)
rsigc["NNLO", "gg"] up to O(ep)
rsigc["NNLO", "qb"] up to O(ep)
rsigc["NNLO", "qg"] up to O(ep)
rsigc["NNLO", "qp"] up to O(ep)
rsigc["NNLO", "qq"] up to O(ep)
Notation:
- "qb" corresponds to "q\bar{q}"
- "qp" corresponds to "qq^\prime"
- PlusDistribution[i,1-x] is the generalized function
(Log[1-x]^i/(1-x))_+, regularized at x = 1.
By definition, PlusDistribution[-1,1-x] = delta(1-x).
- nl is the number of massless quark flavours.
In the Standard Model we have nl=5.
results_conv/
=============
contains the analytical results of the convolutions
listed in Tables 2 and 3 of the paper.
splitfns.m
==========
contains the splitting functions entering our calculation
as extracted from
S. Moch, J.A.M. Vermaseren and A. Vogt, Nucl.\ Phys.\ B {\bf 688} (2004) 101
A. Vogt, S. Moch and J.A.M. Vermaseren, Nucl.\ Phys.\ B {\bf 691} (2004) 129
as a replacement list like
splitfns = {
Pqq1 -> ...,
Pqq2 -> ...,
Pqg1 -> ...,
Pqg2 -> ...,
Pgq1 -> ...,
Pgq2 -> ...,
Pgq3 -> ...,
Pgg1 -> ...,
Pgg2 -> ...,
Pgg3 -> ...
}
where the number refers to the loop order. More information on the
splitting functions is given in Section 2 of the paper.
Note that in the above references the results for the splitting
functions do not contain +-distributions, rather one finds the
statement "Divergences for x->1 are understood in the sense of
+-distributions." which we interpreted as:
(1) Apply HPLLogExtract[] to extract the singularities for
x->1.
(2) If a term contains HPL[{0}, x] = Log[x], it is not a
+-distribution.
(3) If a term contains HPL[{1}, x] = - Log[1-x], this logarithm
together with the denominator 1/(1-x) is considered as a
+-distribution. Other HPLs are left as they are.
(4) If the term contains neither HPL[{0}, x] nor HPL[{1}, x],
1/(1-x) is considered as a +-distribution and other HPLs are
left as they are.