DETERMINING K FACTORS
FOR THE DRELL-YAN
PROCESS AT THE MC
LEVEL
Rob Hickling
MOTIVATION
  At
low mass high order contributions can cause a
deviation in the double differential cross section
of up to 30%.
  Calculating NLO and NNLO cross section can be
computationally time consuming (a single NNLO
calculation can take up to an hour*).
  More efficient to create a parametrisation that
corrects the LO calculation.
  This parametrisation is the K factor.
* This is for one Q value integrating over 20 rapidity bins
K Factor Definitions
  For
this study I have defined the k factors as
follows.
d 2σ
d 2σ
= K NLO
dmdy NLO
dmdy LO
€
€
2
d 2σ
d
σ
1
= K NNLO
dmdy NNLO
dmdy NLO
d 2σ
d 2σ
= K NNLO
dmdy NNLO
dmdy LO
STUDY GOAL
  What
we need is a function that returns a K factor for
a given mass.
  To
calculate the double differential cross sections I will
be using Lance Dixons program Vrap.
  The
double differential cross sections were calculated
first at zero rapidity and then over the whole range.
  A standard 0.1% error was given so fits could be
applied.
RESULTS KNLO(NLO/LO)
 
MRST2007lomod and MSTW2008nlo68cl pdfs
 
Least squares fit in root
 
3rd order polynomial fit
 
The fit is good to 0.3% over the whole data range.
zero rapidity‫‏‬
RESULTS K1NNLO(NNLO/NLO)‫ ‏‬zero rapidity
 
MSTW2008nlo68cl and MSTW2008nnlo68cl pdfs
 
Least squares fit in root
 
4th order polynomial fit
 
The fit is again good to 0.3% over most of the range with a slightly larger
deviation going out to 0.7% at low mass.
RESULTS KNNLO(NNLO/LO)‫ ‏‬zero rapidity
 
MRST2007lomod and MSTW2008nnlo68cl pdfs
 
Least squares fit in root
 
4th order polynomial fit
 
The fit picks up the discrepancy from the NNLO part at low mass but still
has better than 1% agreement over the whole range.
Integrated Over All Y
  I
created a function that uses Simpson rule to
numerically integrate the ddcs over the full y
range (0 < x < 1).
  There is some kind of bug in the program giving
a few unrealistic values, so I have taken these
out leaving some holes in the data
RESULTS KNLO(NLO/LO)‫‏‬
 
MRST2007lomod and MSTW2008nlo68cl pdfs
 
Least squares fit in root
 
5th order polynomial fit
 
The fit is much less accurate over the whole y range. Good to around 1.5%
RESULTS K1NNLO(NNLO/NLO)‫‏‬
 
MSTW2008nlo68cl and MSTW2008nnlo68cl pdfs
 
Least squares fit in root
 
3rd order polynomial fit
 
This is comparing 2 MSTW pdfs so the NNLO NLO difference is smoother
leading to a better fit good to 0.5%.
RESULTS KNNLO(NNLO/LO)‫‏‬
 
MRST2007lomod and MSTW2008nnlo68cl pdfs
 
Least squares fit in root
 
6th order polynomial fit
 
The combined NLO and NNLO corrections actually give quite a flat k factor
parametrisation of around 1.15 with the fit becoming inaccurate below 20
GeV
CONCLUSIONS AND FUTURE WORK
  The
k factors at zero rapidity have a greater correction
than those over the whole y range.
  This
means that the k factor is dependant on rapidity,
which needs to be considered when applying this to
data.