-
NNPDF3.0: PARTON DISTRIBUTIONS FOR
THE LHC RUN IIChristopher S. Deans (University of Edinburgh)
On behalf of the NNPDF collaboration:R. D. Ball, V. Bertone, S.
Carrazza, CSD, L. Del Debbio, S. Forte, P. Groth-Merrild, A.
Guffanti, N. P. Hartland, Z. Kassabov, J. I. Lattore, J. Rojo, L.
Rottoli, Maria Ubiali
28th April 2015
DIS2015 Dallas, Texas
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Christopher S. Deans DIS2015, 28/04/15
OUTLINE
• NNPDF3.0: - Methodology- Data- Closure testing- Results
• Recent work: - PDFs with threshold resummation- Towards
NNPDF3.1
• Summary
2
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Christopher S. Deans DIS2015, 28/04/15
• New PDF set released last November
• arXiv:1410.8849, JHEP 04(2015)040
• LO, NLO and NNLO sets with different values of and different
datasets are available from LHAPDF (lhapdf.hepforge.org)
NNPDF3.0ar
Xiv
:141
0.88
49v3
[he
p-ph
] 4
Mar
201
5
NNPDF Edinburgh 2014/15IFUM-1034-FTCERN-PH-TH/2013-253
OUTP-14-11pCAVENDISH-HEP-14-11
Parton distributions for the LHC Run II
The NNPDF Collaboration:Richard D. Ball1,2, Valerio Bertone2,
Stefano Carrazza4,2,
Christopher S. Deans1, Luigi Del Debbio1, Stefano Forte4,
Alberto Guffanti5,Nathan P. Hartland1, José I. Latorre3, Juan
Rojo6 and Maria Ubiali7
1 The Higgs Centre for Theoretical Physics, University of
Edinburgh,JCMB, KB, Mayfield Rd, Edinburgh EH9 3JZ, Scotland
2 PH Department, TH Unit, CERN,CH-1211 Geneva 23,
Switzerland
3 Departament d’Estructura i Constituents de la Matèria,
Universitat de Barcelona,Diagonal 647, E-08028 Barcelona, Spain
4 Dipartimento di Fisica, Università di Milano and INFN,
Sezione di Milano,Via Celoria 16, I-20133 Milano, Italy
5 Niels Bohr International Academy and Discovery Center,Niels
Bohr Institute, University of Copenhagen,Blegdamsvej 17, DK-2100
Copenhagen, Denmark
6 Rudolf Peierls Centre for Theoretical Physics, 1 Keble
Road,University of Oxford, OX1 3NP Oxford, UK
7 The Cavendish Laboratory, University of Cambridge,J.J. Thomson
Avenue, CB3 0HE, UK
Abstract
We present NNPDF3.0, the first set of parton distribution
functions (PDFs) determined witha methodology validated by a
closure test. NNPDF3.0 uses a global dataset including
HERA-IIdeep-inelastic inclusive cross-sections, the combined HERA
charm data, jet production fromATLAS and CMS, vector boson rapidity
and transverse momentum distributions from ATLAS,CMS and LHCb, W+c
data from CMS and top quark pair production total cross sections
fromATLAS and CMS. Results are based on LO, NLO and NNLO QCD theory
and also includeelectroweak corrections. To validate our
methodology, we show that PDFs determined frompseudo-data generated
from a known underlying law correctly reproduce the statistical
distribu-tions expected on the basis of the assumed experimental
uncertainties. This closure test ensuresthat our methodological
uncertainties are negligible in comparison to the generic
theoretical andexperimental uncertainties of PDF determination.
This enables us to determine with confidencePDFs at different
perturbative orders and using a variety of experimental datasets
ranging fromHERA-only up to a global set including the latest LHC
results, all using precisely the samevalidated methodology. We
explore some of the phenomenological implications of our results
forthe upcoming 13 TeV Run of the LHC, in particular for Higgs
production cross-sections.
1
3
http://lhapdf.hepforge.org
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Christopher S. Deans DIS2015, 28/04/15
NNPDF APPROACH
• Parameterise parton distributions using neural networks
• Perform a genetic algorithm minimisation to find best-fit
PDFs
• Cross-validation to prevent neural networks from
over-learning
• Capture PDF uncertainties by generating set of Monte Carlo
replica PDFs
• Fit using large, global dataset, including data from fixed
target DIS and DY, HERA, Tevatron and LHC
4
• New code completely rewritten in C++, with new structure to
improve efficiency and promote modularity between theory and
fitting.
• New mutation strategy exploiting the structure of the neural
networks to obtain a better fit results in a shorted amount of
time.
• Improved cross-validation which prevents the fit from stopping
too early while still protecting against over-learning.
• New positivity constraints covering a wider range of
observables over a larger kinematic range.
3.0 IMPROVEMENTS
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Christopher S. Deans DIS2015, 28/04/15
x-6
10-5
10 -410-3
10 -210 -110 1
]2
[ G
eV
2 T /
p2
/ M
2Q
1
10
210
310
410
510
610
710FT DIS
HERA1
FT DY
TEV EW
TEV JET
ATLAS EW
LHCB EW
LHC JETS
HERA2
ATLAS JETS 2.76TEV
ATLAS HIGH MASS
ATLAS WpT
CMS W ASY
CMS JETS
CMS WC TOT
CMS WC RAT
LHCB Z
TTBAR
NNPDF3.0 NLO dataset
NNPDF3.0: DATASET
• HERA: HERA-II structure function data from H1 & Zeus,
combined charm production data
• ATLAS: 2.76 TeV inclusive jet data, high-mass Drell-Yan and W
pT distributions, top quark pair production total cross-section
• CMS: 7 TeV inclusive jet data, muon asymmetry, W+c production,
double differential Drell-Yan distributions, top quark pair
production cross-section
• LHCb: Z rapidity distribution
Large amount of new data included in the fit:
5
… plus data from NNPDF2.3
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Christopher S. Deans DIS2015, 28/04/15
-410
-310
-210
-110
1
10
210
(GeV)T
p200 400 600 800 1000 1200 1400 1600 1800 2000
|y|
0
0.5
1
1.5
2
2.5
| for gg-channelδCMS 2011 - | | for gg-channelδCMS 2011 - |
NNPDF3.0: THEORY
• Use fast interfaces at NLO (APPLgrid, FastNLO, aMCfast) to
generate FKtables
• For NNLO fits, supplement NLO theory for hadron data with
k-factors
• Jet data included at NNLO using threshold calculation in
regions where the gg channel agrees with exact calculation (Pires
et al.)
• Electroweak corrections from FEWZ3.1 for neutral current
datasets included as additional factor
6
Percentage difference between approximate and exact calculation
for gg channel
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Christopher S. Deans DIS2015, 28/04/157
NNPDF3.0: CLOSURE TESTINGValidate methodology by fitting to
pseudo-data generated using known PDFs
Generate pseudo-data using underlying law and experimental
covariance matrix
Fit pseudo-data using standard fitting methodology
Compare results obtained from fit to underlying law
Define underlying law based on chosen PDF set (MSTW, NNPDF, CT
etc.)
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Christopher S. Deans DIS2015, 28/04/15
NNPDF3.0: CLOSURE TEST RESULTS• Reproduce central of input set
to within 1% (Fit
=1.00, input =1.01)
• Additionally reproduce of individual datasets (see right)
• PDF central values reproduced (with expected fluctuations)
• Theory within one sigma of fit central value in ~70% of
fits.
8
Experiments
NMC SLACBCDMS
CHORUSNTVDMN
HERA1AV
ZEUSHERA2
H1HERA2
HERAF2CHARM
DYE886DYE605
CDF D0 ATLASCMS LHCB
TOP
2χ
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
for experiments2χDistribution of
2χClosure test 2χMSTW2008nlo
2χClosure test central 2χMSTW2008nlo central
for experiments2χDistribution of
)σDifference to theory (-5 -4 -3 -2 -1 0 1 2 3 4 5
Entri
es
020406080
100120140160180200 Replica distribution
Gaussian distribution
Distribution of single replica fits in level 2 uncertainties
x5−10 4−10 3−10 2−10 1−10 1
)2xg
(x,Q
2−1−
01
2
34
5
67
Closure test fit to MSTW2008 pseudo-data
Closure test fit
MSTW2008
Closure test fit to MSTW2008 pseudo-data
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Christopher S. Deans DIS2015, 28/04/159
x5−10 4−10 3−10 2−10 1−10 1
)2xg
(x,Q
2−1−
01
2
34
5
67
Closure test fit to NNPDF2.3 pseudo-data
Closure test fit
NNPDF2.3
Closure test fit to NNPDF2.3 pseudo-data
CLOSURE TEST PDFS
x5−10 4−10 3−10 2−10 1−10 1
)2xg
(x,Q
2−1−
01
2
34
5
67
Closure test fit to NNPDF3.0 pseudo-data
Closure test fit
NNPDF3.0
Closure test fit to NNPDF3.0 pseudo-data
x5−10 4−10 3−10 2−10 1−10 1
)2xg
(x,Q
2−1−
01
2
34
5
67
Closure test fit to CT10 pseudo-data
Closure test fit
CT10
Closure test fit to CT10 pseudo-data
x5−10 4−10 3−10 2−10 1−10 1
)2xg
(x,Q
2−1−
01
2
34
5
67
Closure test fit to MSTW2008 pseudo-data
Closure test fit
MSTW2008
Closure test fit to MSTW2008 pseudo-data
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Christopher S. Deans DIS2015, 28/04/15
Number of Generations310 410 510
2χ
-410
-310
-210
-110Old (2.3) genetic algorithm
New genetic algorithm
Effectiveness of Genetic Algorithm in Level 0 Closure Tests
NNPDF3.0: NOISELESS CLOSURE TEST• When generating closure
test
pseudo-data, it’s possible to control level of noise
• Closure test with no noise provides test of basic fitting
ability
• Ideal of zero (expect slightly above for fixed length fit)
10
x-510 -410 -310 -210 -110 1
)2
xg(x
,Q
-2
-1
0
1
2
3
4
5
6
7
Level 0 closure test vs. MSTW
Level 0 Closure Test Fit
MSTW2008nlo68cl
Level 0 closure test vs. MSTW
x-510 -410 -310 -210 -110 1
)2
(x,Q
ux
0
0.2
0.4
0.6
0.8
1
1.2
Level 0 closure test vs. MSTW
Level 0 Closure Test Fit
MSTW2008nlo68cl
Level 0 closure test vs. MSTW
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Christopher S. Deans DIS2015, 28/04/15
NNPDF3.0: RESULTS
11
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Christopher S. Deans DIS2015, 28/04/1512
x -510 -410 -310 -210 -110
)2 (
x, Q
+x
s
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.72 = 2 GeV2 = 0.118, QSαNNPDF3.0 NLO,
NNPDF3.0
NNPDF2.3
2 = 2 GeV2 = 0.118, QSαNNPDF3.0 NLO,
NNPDF3.0 RESULTS: PDFS
x -510 -410 -310 -210 -110
)2x
g ( x
, Q
-1
0
1
2
3
4
52 = 2 GeV2 = 0.118, QSαNNLO,
NNPDF3.0
NNPDF2.3
2 = 2 GeV2 = 0.118, QSαNNLO,
x -510 -410 -310 -210 -110
)2 (
x, Q
Σx
00.5
11.5
22.5
33.5
44.5
52 = 2 GeV2 = 0.118, QSαNNLO,
NNPDF3.0
NNPDF2.3
2 = 2 GeV2 = 0.118, QSαNNLO,
• General agreement with 2.3; differences of ~1σ due to new
data/improved methodology
• Reduction of uncertainties in many PDFs
• Impact of extended positivity in s+
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Christopher S. Deans DIS2015, 28/04/1513
( GeV )XM10210 310
Qua
rk -
Qua
rk L
umin
osity
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
)=0.118 - Ratio to NNPDF3.0Z
(MSαLHC 13 TeV, NNLO,
NNPDF3.0CT10MMHT14
)=0.118 - Ratio to NNPDF3.0Z
(MSαLHC 13 TeV, NNLO,
PDF LUMINOSITIES
( GeV )XM10210 310
Qua
rk -
Antiq
uark
Lum
inos
ity
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
)=0.118 - Ratio to NNPDF3.0Z
(MSαLHC 13 TeV, NNLO,
NNPDF3.0CT10MMHT14
)=0.118 - Ratio to NNPDF3.0Z
(MSαLHC 13 TeV, NNLO,
( GeV )XM10210 310
Qua
rk -
Glu
on L
umin
osity
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
)=0.118 - Ratio to NNPDF3.0Z
(MSαLHC 13 TeV, NNLO,
NNPDF3.0CT10MMHT14
)=0.118 - Ratio to NNPDF3.0Z
(MSαLHC 13 TeV, NNLO,
( GeV )XM10210 310
Glu
on -
Glu
on L
umin
osity
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
)=0.118 - Ratio to NNPDF3.0Z
(MSαLHC 13 TeV, NNLO,
NNPDF3.0CT10MMHT14
)=0.118 - Ratio to NNPDF3.0Z
(MSαLHC 13 TeV, NNLO,
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Christopher S. Deans DIS2015, 28/04/1514
x -410 -310 -210 -110
) [re
f] 2
) [ne
w] /
V (
x, Q
2V
( x, Q
0.85
0.9
0.951
1.05
1.1
1.151.2
1.25
1.32 GeV4 = 102 = 0.118, QSαNNLO,
Global Fit
No LHC data
2 GeV4 = 102 = 0.118, QSαNNLO,
IMPACT OF LHC DATA
x -410 -310 -210 -110
) [re
f] 2
( x,
Q+
) [ne
w] /
s2
( x,
Q+ s 0.8
0.9
1
1.1
1.2
1.3
2 GeV4 = 102 = 0.118, QSαNNLO,
Global Fit
No LHC data
2 GeV4 = 102 = 0.118, QSαNNLO,
x -410 -310 -210 -110
) [re
f] 2
) [ne
w] /
g (
x, Q
2g
( x, Q
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.252 GeV4 = 102 = 0.118, QSαNNLO,
Global Fit
No LHC data
2 GeV4 = 102 = 0.118, QSαNNLO, • Compare NNPDF3.0 fits with and
without LHC data
• Reduction of uncertainties in large-x gluon, light-quarks
• Also available on LHAPDF: HERA-only, HERA+ATLAS, HERA+CMS,
NoJet PDFs
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Christopher S. Deans DIS2015, 28/04/1515
x -510 -410 -310 -210 -110
)2x
g ( x
, Q
-1
0
1
2
3
4
52 = 2 GeV2 = 0.118, QSαNNLO,
Global
=1.1maxαConservative,
=1.2maxαConservative,
2 = 2 GeV2 = 0.118, QSαNNLO,
CONSERVATIVE DATASET PDFS• PDFs generated by fitting reduced
fully
consistent dataset
• Consistency determined by calculating P( ) for each dataset,
(see arXiv:1012:0836)
• Results compatible with global fit, with larger
uncertainties
αmax = 1.1 αmax = 1.2 αmax = 1.3 Global fitχ2nlo χ
2nnlo χ
2nlo χ
2nnlo χ
2nlo χ
2nnlo χ
2nlo χ
2nnlo
Total 0.96 1.01 1.06 1.10 1.12 1.16 1.23 1.29NMC d/p 0.91 0.91
0.89 0.89 0.88 0.89 0.92 0.93
NMC σNC,p - - - - - - 1.63 1.52SLAC - - - - 1.77 1.19 1.59
1.13
BCDMS - - 1.11 1.15 1.12 1.16 1.22 1.29CHORUS - - 1.06 1.02 1.09
1.07 1.11 1.09NuTeV 0.35 0.34 0.62 0.64 0.70 0.70 0.70 0.86HERA-I
0.97 0.98 1.02 1.00 1.02 0.99 1.05 1.04
ZEUS HERA-II - - - - 1.41 1.48 1.40 1.48H1 HERA-II - - - - - -
1.65 1.79HERA σcNC - - 1.21 1.32 1.20 1.31 1.27 1.28E886 d/p 0.30
0.30 0.43 0.40 0.44 0.46 0.53 0.48E886 p - - 1.18 1.40 1.27 1.53
1.19 1.55E605 1.04 1.10 0.74 0.83 0.75 0.88 0.78 0.90
CDF Z rapidity - - - - - - 1.33 1.53CDF Run-II kt jets - - 1.01
2.01 1.04 1.84 0.96 1.80
D0 Z rapidity 0.56 0.61 0.62 0.71 0.60 0.69 0.57 0.61ATLAS W,Z
2010 - - 1.19 1.13 1.19 1.17 1.19 1.23
ATLAS 7 TeV jets 2010 0.96 1.65 1.08 1.58 1.10 1.54 1.07
1.36ATLAS 2.76 TeV jets 1.03 0.38 1.38 0.36 1.35 0.35 1.29
0.33ATLAS high-mass DY - - - - - - 2.06 1.45
ATLAS W pT - - - - - - 1.13 -CMS W electron asy 0.98 0.84 0.82
0.72 0.85 0.73 0.87 0.73CMS W muon asy - - - - - - 1.81 1.72CMS
jets 2011 0.90 2.09 0.96 2.09 0.99 2.10 0.96 1.90
CMS W + c total - - - - - - 0.96 0.84CMS W + c ratio - - - - - -
2.02 1.77CMS 2D DY 2011 - - - - 1.20 1.30 1.23 1.36LHCb W rapidity
- - 0.69 0.65 0.74 0.69 0.71 0.72LHCb Z rapidity - - 1.23 1.78 1.11
1.58 1.10 1.59
σ(tt̄) - - - - - - 1.43 0.66
Table 11: The experimental χ2 values at NLO and NNLO for
NNPDF3.0 fits to conservative datasetscorresponding to three
different values of the threshold αmax (see text). In each case,
the χ2 is shown forthe datasets which pass the conservative cut.
The values for the global fit (same as in Tab. 9) are alsoshown for
ease of comparison.
datasets such as the H1 HERA-II data, the ATLAS high-mass
Drell-Yan data, and the CMSW+c data are excluded even from the
least conservative set, the one with αmax = 1.3. On theother hand,
the CMS inclusive jet data is included for all values of αmax; note
that for thisdataset the experimental χ2 shown in Tab. 11 is
significantly worse than the t0 value used forminimization and the
determination of P (α).
The maximally consistent dataset, found with αmax = 1.1,
includes the NMC d/p data, theNuTeV and HERA-I DIS data, the
Drell-Yan data from E866 and E605, the D0 Z rapidity, theATLAS and
CMS inclusive jets and the CMS W electron asymmetry.
In Tab. 12, we furthermore compare the mean, mode and median of
the P (α) distributionsfor the experiments excluded from the NNLO
conservative fit with αmax = 1.1 when the globalfit is used as
prior (i.e. the same numbers for the corresponding entries in Tab.
10), to thesame quantities computed using as a prior the
conservative fit itself. All the peak values ofP (α) deteriorate
when using the conservative set as a prior, as they ought to.
Clearly, thisdeterioration will be maximal for datasets which are
internally consistent, but inconsistent with
91
-
Christopher S. Deans DIS2015, 28/04/1516
• NNPDF3.0 fit including deuteron corrections to fixed-target
DIS and DY data (Thorne et al., arXiv:1412.3989)
• Non-negligible difference only for down PDF
• CMS W+c and ATLAS WZ data provide information on strangeness
content
• Fits with excluding different sets of relevant data find
general agreement
• Shown:
x 2−10 1−10
) [re
f] 2
) [ne
w] /
d (
x, Q
2d
( x, Q
0.9
0.95
1
1.05
1.1
1.15
1.2
2 GeV4 = 102 = 0.118, QSαNNPDF3.0 NLO,
No deuteron corrections
MMHT14 deuteron corrections
2 GeV4 = 102 = 0.118, QSαNNPDF3.0 NLO,
x -310 -210 -110
00.20.40.60.8
11.21.41.61.8
22 = 2 GeV2 = 0.118, QSαNNLO,
Global Fit
no Neutrino Data
no W+c Data
no Neutrino, W+c Data
2 = 2 GeV2 = 0.118, QSαNNLO,
rs =s+ s̄
ū+ d̄
rs =s+ s̄
ū+ d̄(1)
1
DEUTERON CORRECTIONS
PROTON STRANGENESS
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Christopher S. Deans DIS2015, 28/04/15
LHC CROSS-SECTIONS
• Consistent results between 2.3 and 3.0 across many LHC
observables
• Slightly larger uncertainties in general: due to improvements
in methodology, better exploration of parameter space
• Decrease in ggH cross-section due to softer gg luminosity
• Again change is due to improved methodology, similar results
for 3.0 global and 2.3data fits
17
( gg
> H
) [p
b]σ
43
44
45
46
47
=0.118SαNNPDF3.0 NNLO, LHC 13 TeV, iHixs1.3.3,
NNPDF2.32.3data
GlobalConservative
noJetsnoLHC
HERAonly
=0.118SαNNPDF3.0 NNLO, LHC 13 TeV, iHixs1.3.3,
Ratio to NNPDF2.30.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1
1.12 1.14
=0.118, MadGraph5_aMC@NLO fNLOSαLHC 13 TeV,
- e+ -> e0p + p -> Zeν + -> e
+p + p -> W
eν - -> e-p + p -> W
tp + p -> t
tp + p -> H t
c -p + p -> W
+ jetγp + p ->
eν +p + p -> H e
- e+p + p -> H e
+ jeteν +p + p -> e
NNPDF3.0NNPDF2.3
NNPDF3.0 cons
=0.118, MadGraph5_aMC@NLO fNLOSαLHC 13 TeV,
-
Christopher S. Deans DIS2015, 28/04/15
RECENT WORK
18
-
Christopher S. Deans DIS2015, 28/04/1519 In the NLO+NLL fit,
effects of resummation can be large, up to -10% (for quarks) and
+30% (for gluons) at high-x, relevant for the production of new BSM
particles!!
5
PDFs with threshold resummation Effects of threshold resummation
most relevant for large-x DIS structure functions and for
Drell-Yan
forward rapidity distributions! Threshold resummation not
available for inclusive jets, so compare with DIS+DY+top NNPDF3.0
baseline!
!
Juan Rojo PDF4LHC, CERN, 13/04/2015
THRESHOLD RESUMMATION• First PDF sets at NLO+NLL and
NNLO+NNLL
• Soft-gluon resummation most relevant for DIS at large-x and
Drell-Yan at forward rapidities
• Perform fit (plus baseline) with DIS, DY and top data — code
not yet available for inclusive jet processes
• Preliminary results for NLO+NLL: shifts in quark and gluon at
large-x, relevant for production of BSM particles
PRELI
MINAR
Y
PRELI
MINAR
Y
-
Christopher S. Deans DIS2015, 28/04/15
x-510 -410 -310 -210 -110 1
Rat
io
0.6
0.8
1
1.2
1.4
1.6xu(x,Q), comparison plot
NNPDF30 NLO
NNPDF30 NLO + RW(LHCb W)
Q = 1.41 GeV
Gen
erat
ed b
y A
PFEL
2.0.
1: V
.Ber
tone
, S.C
arra
zza,
J.R
ojo
(arX
iv:1
310.
1394
)
TOWARDS NNPDF3.1
• New Data: Final combined HERA results, Tevatron WZ, many new
LHC processes incl. ATLAS 1,2,3-jet cross-sections, ATLAS top
differential distribution, LHCb W, Z rapidity, CMS 8TeV Drell-Yan,
ATLAS prompt photon
• Running quark mass effects
• Fitted charm PDF: parameterisation with 9 independent
flavors
• PDF evolution and DIS cross-sections calculated in APFEL;
optimised theory calculation in fit
20
NNPDF3.0 reweighted with LHCb W data
-
Christopher S. Deans DIS2015, 28/04/15
SUMMARY
• NNPDF3.0: - Next generation PDF set, available now on LHAPDF-
Large amount of new data from HERA, ATLAS, CMS and LHCb- New
theory: Electroweak corrections, top cross-sections, jet data at
NNLO- Improved fitting methodology fully validated using closure
tests
• Looking ahead: - First PDF set including NLL and NNLL
soft-gluon resummation effects- NNPDF3.1: More LHC data, HERA
combined dataset, running quark masses, Fitted charm PDF
• Also available: - NNPDF2.3QED: PDFs with QED evolution +
photon, arXiv:1308.0598- NNPDFpol1.1: Polarised neural network
PDFs, arXiv:1406.5539
21