MT: a Mathematica package to compute convolutions
by
Maik H\"oschele, Jens Hoff, Alexey Pak, Matthias Steinhauser, Takahiro Ueda
arXiv:1307.6925, SFB/CPP-13-50, TTP13-27, LPN13-052
Information is provided about the Mathematica package MT
presented and the analytical results obtained/used in
the above paper.
sig_DY_LO_NLO_NNLO.m
====================
contains analytical results for the partonic Drell-Yan cross sections
up to NNLO expressed in terms of HPLs and plus distributions
(see Table 2 of the paper), which have been computed up to an
order in the dimensional regulator ep, which is sufficiently high for
an N3LO calculation:
rsigc["LO", "qb"] to all orders in ep
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:
(see Table 3 and 4 of the paper)
- "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.
- CF, CA, TF color factors
In QCD we have CF = 4/3, CA = 3, TF = 1/2
- Lv is Log[mu^2/Q^2]
coeff[] collects various coefficients that contain
- alphas is the strong coupling constant
- Sum[k,QB] is the sum over k element quarks Q and anti-quarks B
- qi is the quark q with index i
- bi is the aniquark b with in dex i
- Cii[qi,bj] is the coupling matrix to vector bosons with superscripts ii
- Cif[qi,qj] is the coupling matrix to vector bosons with superscripts if
- Cff[qk,bl] is the coupling matrix to vector bosons with superscripts ff
- vi vector coupling
- ai axial vector coupling
- delta[i,j] is the Kronecker delta
splitfnsMVV.m
==========
contains the splitting functions 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
splitfnsMVV = {
Pqg1 -> ...,
Pqg2 -> ...,
Pqg3 -> ...,
Pgq1 -> ...,
Pgq2 -> ...,
Pgq3 -> ...,
Pgg1 -> ...,
Pgg2 -> ...,
Pgg3 -> ...,
Pqq1 -> ...,
Pnsp2 -> ...,
Pnsp3 -> ...,
Pnsm2 -> ...,
Pnsm3 -> ...,
Pps2 -> ...,
Pps3 -> ...
}
where the number refers to the loop order. More information on the
splitting functions is given in Table 1 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.
MT-1.0.tar.gz
=============
contains MT package together with additional material including a
README file which in detail explains installation etc.
Usage and examples are provided in example.nb
(see Sections 3, 4 and 5 of the paper as well).
MTtable1.tar.gz, MTtable2.tar.gz,... MTtable8.tar.gz
====================================================
contains tables
MTtab1.m, MTtab2.m, ... MTtab8.m for Mellin transforms
and tables
MTitab1.m, MTitab2.m, ... MTitab8.m for inverse Mellin transforms
up to weight 1, 2, ... 8 of the HPLs.