NNPDF1.0: parton distribution funtions with faithful errors
Luigi Del Debbio
luigi.del.debbio@ed.ac.uk
University of Edinburgh
arXiv:0808.1231 [hep-ph], Nucl. Phys. B (in press)
The NNPDF Collaboration
R.D.Ball1 , L.Del Debbio1 , S.Forte2 , A.Guffanti3 , J.I.Latorre4 , A. Piccione2 , J. Rojo2 , M.U.1
1
PPT Group, School of Physics, University of Edinburgh
2
3
4
Dipartimento di Fisica, Università di Milano
Physikalisches Institut, Albert-Ludwigs-Universität Freiburg
Departament d’Estructura i Constituents de la Matèria, Universitat de Barcelona
nnpdf1.0, taiwan 08 – p.1/41
DIS Parton distribution functions
Deep inelastic observables:
2
FI (x, Q ) =
X
j
γ∗, W ∗, Z∗
CIj (x, αs (Q2 )) ⊗ fj (x, Q2 )
q
Scale–dependence of PDFs:
∂
Q
fi (x, Q2 )
2
∂Q
2
fi (x, Q2 )
=
X
j
=
X
j
Back to the observables:
2
FI (x, Q )
Pij (x, αs (Q2 )) ⊗ fj (x, Q2 )
p
X
Γij (x, αs , α0s ) ⊗ fj (x, Q20 )
=
X
jk
=
X
j
CIj (x, αs ) ⊗ Γjk (x, αs , α0s ) ⊗ fk (x, Q20 )
KIj (x, αs , α0s ) ⊗ fj (x, Q20 )
nnpdf1.0, taiwan 08 – p.2/41
Parton distributions for LHC
γ∗, W ∗, Z∗
p′
q̄
q
p
⇒
X
•
•
•
q
H
W, Z
p
different kinematics
nonperturbative nucleon structure described by the same PDFs
evolved to the relevant scales
nnpdf1.0, taiwan 08 – p.3/41
LHC kinematics
LHC parton kinematics
9
10
8
10
x1,2 = (M/14 TeV) exp(±y)
Q=M
M = 10 TeV
7
10
6
M = 1 TeV
5
10
4
M = 100 GeV
10
2
2
Q (GeV )
10
•
•
•
PDFs need to be extrapolated
•
important for modelling the QCD
background at LHC
•
necessity to develop PDFs with faithful errors
uncertainty in the extrapolation region
uncertainty propagates into LHC physical
observables
3
10
y=
6
4
2
0
2
4
6
2
10
M = 10 GeV
1
fixed
target
HERA
10
0
10
-7
10
-6
10
-5
10
-4
-3
10
10
-2
10
-1
10
0
10
x
nnpdf1.0, taiwan 08 – p.4/41
Partons with errors
x=6.32E-5 x=0.000102
x=0.000161
x=0.000253
x=0.0004
x=0.0005
x=0.000632
x=0.0008
5
given a set of data points, determine a set of
functions with errors
ZEUS NLO QCD fit
H1 PDF 2000 fit
H1 94-00
H1 (prel.) 99/00
x=0.0013
ZEUS 96/97
BCDMS
x=0.0021
E665
4
x=0.0032
NMC
data included in CTEQ5 parton fit
x=0.005
x=0.008
3
DIS (fixed target)
HERA (’94)
DY
W-asymmetry
Direct-γ
Jets
x=0.013
102
x=0.021
x=0.032
2
Q (GeV)
em
F2 -log10(x)
HERA F2
x=0.05
x=0.08
x=0.13
101
x=0.18
1
x=0.25
x=0.4
x=0.65
0
1
10
10
2
10
3
10
4
2
10
2
Q (GeV )
5
100
100
101
102
103
104
1/X
nnpdf1.0, taiwan 08 – p.5/41
What’s the problem?
•
•
•
[Kosower 99]
for a single quantity, we quote 1 sigma errors: value ± error
for a pair of numbers, we quote a 1 sigma ellipse
for a function, we need an “error bar” in a space of functions
we must determine the probability density (measure) P[fi (x)] in the space of parton
distribution functions fi (x) (i=quark, antiquark, gluon)
EXPECTATION VALUE OF F [fi (x)] ⇒ FUNCTIONAL INTEGRAL
D
F [fi (x)]
E
=
Z
Dfi F [fi (x)] P[fi ],
we must extract from the data a description of the probability distribution P
nnpdf1.0, taiwan 08 – p.6/41
The standard solution
•
choose a parameterization at a reference scale
•
evolve to desired scale & compute physical observables
•
determine best-fit values of parameters
•
determine error by propagation of error on parameters (’hessian method’)
or by parameter scans (’lagrange multiplier method’)
problem projected onto the finite–dimensional space of parameters
nnpdf1.0, taiwan 08 – p.7/41
Comparing global fits
W production cross-section Tevatron
PDF set
xsec [nb]
PDF uncertainty
Alekhin
2.73
MRST2002
2.59
± 0.05
CTEQ6
2.54
Higgs production at LHC
4
1.15
1.1
1.05
1
0.95
0.9
100
± 0.03
± 0.10
[Thorne 03]
Alekhin vs. MRST/CTEQ → W production
xsect at tevatron do not agree within respective errors
Alekhin vs. MRST/CTEQ → predictions for
associate Higgs W production LHC do not
agree within respective errors
2
150
200
1
MRST
CTEQ
Alekhin
0.5
σ(pp → HW ) [pb]
√
s = 14 TeV
0.3
100
120
140
160
MH [GeV]
180
200
[Djouadi and Ferrag 04]
nnpdf1.0, taiwan 08 – p.8/41
Troubles with error bars
PDF4LHC workshop at CERN 08
benchmark fits on reduced sets do not agree with global fits within errors
incompatible experiments?
lack of generality in the parametrization?
tolerance criterion ∆χ2 > 1?
0.4
MRSTbench
MRST2001
0.3
xdV(x,Q2=20)
•
•
•
•
0.2
0.1
0
10
-3
10
-2
10
-1
x
nnpdf1.0, taiwan 08 – p.9/41
The Neural Monte Carlo
NNPDF collaboration (2004: ldd, Forte, Latorre, Piccione, Rojo; 2007: + Ball, Guffanti, Ubiali)
•
(k)
Monte Carlo replicas FI
(pi ) of the
(data)
(pi )
original dataset FI
⇒ representation of P[FI (pi )] at discrete
set of points pi
•
train a neural net for each pdf on each
replica, → neural representation of the
(net),(k)
pdfs fi
•
The set of neural nets is a representation
of the probability density:
D
F [fi ]
E
=
1
Nrep
Nrep
X
k=1
h
i
(net)(k)
F fi
nnpdf1.0, taiwan 08 – p.10/41
MC sampling of exp data
(art)(k)
FI,p
(k)
(exp)
= Sp,N FI,p
where
(k)
Sp,N
0
@1 +
Nc
X
l=1
(k)
(k)
1
rp,l σp,l + rp σp,s A , k = 1, . . . , Nrep ,
Nr q
Na “
”Y
Y
(k)
(k)
1 + rp,n σp,n
=
1 + rp,n σp,n .
n=1
n=1
nnpdf1.0, taiwan 08 – p.11/41
Monte Carlo errors
•
•
for each replica (k) we fit one set of PDFs
•
statistical properties of any function of the PDFs can be computed using standard
methods:
the ensemble of fitted replicas represents the probability distribution in the space
of PDFs
hF [f (x)]i
=
σF [f (x)]
=
ρ[fa (x1 , Q21 ), fb (x2 , Q22 )]
=
Nrep
1 X
F [f (k)(net) (x)]
Nrep k=1
q
hF [f (x)]2 i − hF [f (x)]i2
hfa (x1 , Q21 )fb (x2 , Q22 )i − hfa (x1 , Q21 )ihfb (x2 , Q22 )i
σa (x1 , Q21 )σb (x2 , Q22 )
nnpdf1.0, taiwan 08 – p.12/41
Experimental data
OBS
Data set
OBS
Data set
F2p
NMC
−
σN
C
ZEUS
SLAC
BCDMS
F2d
F2d /F2p
+
σCC
SLAC
BCDMS
+
σN
C
H1
ZEUS
H1
−
σCC
ZEUS
ZEUS
H1
H1
σν , σν̄
CHORUS
NMC-pd
FL
H1
•
Kinematical cuts:
Q2 >2 GeV2
W 2 = Q2 (1 − x)/x >12.5
GeV2
•
∼ 3000 points.
nnpdf1.0, taiwan 08 – p.13/41
Neural Networks?
each PDF at the reference scale is parametrised by one NN:
fi (x, Q20 ) = Ni (x)
NN is a non–linear mapping:
Ni : Rn → Rm
nnpdf1.0, taiwan 08 – p.14/41
Some details about Neural Networks
Multilayer feed-forward networks
1
•
Each neuron receives input from
neurons in preceding layer and feeds
output to neurons in subsequent layer
•
Activation determined by weights and
thresholds
“P
”
ξi = g
j ωij ξj − θi
0.8
0.6
0.4
•
0.2
-10
-5
5
10
Sigmoid activation function
g(x) = 1+e1−βx
... just another set of basis functions!
eg, a 1-2-1 NN:
(3)
(1)
ξ1 (ξ1 ) =
1
(2)
(2)
ω11
ω12
(3)
1+exp[θ1 −
−
]
(2)
(1) (1)
(2)
(1) (1)
θ1 −ξ1 ω11
θ2 −ξ1 ω21
1+e
1+e
Thm: any function can be represented by a sufficiently big neural network
nnpdf1.0, taiwan 08 – p.15/41
Basis set
•
Each independent PDF at the initial scale Q20 = 2GeV2 is parameterized by an
individual NN.
•
Little constraint on strange → Flavor Assumptions:
- Symmetric strange sea s(x) = s̄(x)
- Strange sea proportional to non-strange sea s̄(x) =
C
¯
(ū(x) + d(x))
2
(C = 0.5)
- Intrinsic heavy quarks contributions neglected.
•
Parametrization of (4+1) combinations of PDFs at Q20 = 2 GeV2 :
Singlet : Σ(x)
7−→ NNΣ (x)
2-5-3-1 37 pars
Gluon : g(x)
7−→ NNg (x)
2-5-3-1 37 pars
Total valence : V (x) ≡ uV (x) + dV (x) 7−→ NNV (x)
2-5-3-1 37 pars
Non-singlet triplet : T3 (x)
7−→ NNT 3 (x)
2-5-3-1 37 pars
¯ − ū(x) 7−→ NN∆ (x)
Sea asymmetry : ∆S (x) ≡ d(x)
2-5-3-1 37 pars
185 parameters
nnpdf1.0, taiwan 08 – p.16/41
Normalization and sum rules
•
•
Σ(x, Q20 )
=
V (x, Q20 )
=
T3 (x, Q20 )
=
∆S (x, Q20 )
=
g(x, Q20 )
=
(1 − x)mΣ x−nΣ NNΣ (x) ,
AV (1 − x)mV x−nV NNV (x) ,
(1 − x)mT3 x−nT3 NNT3 (x) ,
A∆S (1 − x)m∆S x−n∆S NN∆S (x) ,
Ag (1 − x)mg x−ng NNg (x) .
Polynomial Preprocessing → Training Efficiency
Normalization → Fixed by valence and momentum sum rules
Z
1
dx x (Σ(x) + g(x))
=
1
dx (u(x) − ū(x))
=
2
¯
dx (d(x) − d(x))
=
1.
0
Z
1
0
Z
1
0
nnpdf1.0, taiwan 08 – p.17/41
Evolution
Observables are a convolution over x of PDFs and Coefficient Functions:
2
FI (x, Q ) =
X
j
2
CIj (x, αs ) ⊗ fj (x, Q ) =
X
j,k
CIj (x, αs ) ⊗ Γjk (x, αs , α0s ) ⊗ fk (x, Q20 )
nnpdf1.0, taiwan 08 – p.18/41
Kernels for a physical observable
F2 proton structure function
F2p
=
x{
1
5 s
1
1
C2,q ⊗ Σ + C2,q ⊗ (T3 + (T8 − T15 ) + (T24 − T35 ))
18
6
3
5
+he2q iC2,g ⊗ g}
F2p
=
x{KF2,Σ ⊗ Σ0 + KF2,g
„
«
1
⊗ g0 + KF2,+ ⊗ T3,0 + (T8,0 − T15,0 ) }
3
In Mellin space
KF2,Σ
=
KF2,g
=
KF2,+
=
5 s qq
C Γ +
18 2,q S
5 s qg
C Γ +
18 2,q S
1
C2,q Γ+
NS
6
1
35,q
gq
2
C2,q (Γ24,q
−
Γ
iC
Γ
)
+
he
2,g
q
S
S
S
30
1
C2,q (Γ24,g
− Γ35,g
) + he2q iC2,g Γgg
S
S
S
30
nnpdf1.0, taiwan 08 – p.19/41
Theoretical errors
•
•
•
Higher perturbative orders → NLO fit
Heavy quark treatment → Zero Mass Variable Flavor Number scheme.
quarks are radiatively generated at thresholds. [Thorne,Tung, arXiv:0809.0714]
Target Mass Corrections included and factorized into the hard kernels.
2
3
2 F (ξ, Q2 )
M
x
x
2
2
N
+6
I
(ξ,
Q
)
Fe2 (ξ, Q ) = 3/2
2
ξ2
Q2 τ 2
τ
2
2 x2
4MN
τ =1 +
Q2
2x
ξ=
√
1+ τ
Z 1
dz
I2 (ξ, Q2 ) =
F2 (z, Q2 ).
2
ξ z
Taking Mellin transforms with respect to ξ, defines a new target mass corrected
coefficient function
τ 1/2 )2
e2,j (N, αs , τ ) = (1 +
C
4τ 3/2
1+
´!
`
−1/2
3 1−τ
N +1
C2,j (N, αs )
nnpdf1.0, taiwan 08 – p.20/41
Training strategy
•
•
unbiased basis of functions, parametrized by a large number of parameters
•
•
might accomodate statistical fluctutations of the data
•
•
•
•
dynamical stopping by cross validation
genetic algorithms for minimization
optimal training, beyond which the fit is just adjusting to statistical fluctutations
for each replica divide the data randomly into training and validation
minimization performed on the training set only
when the training χ2 still decreases while the validation χ2 stops decreasing
→ STOP
nnpdf1.0, taiwan 08 – p.21/41
Dynamical stopping
2.8
Training Dataset
2.7
Validation Dataset
Stopping point
E
2.6
2.5
2.4
2.3
2.2
500
1000
1500
2000
Iterations
2500
3000
3500
nnpdf1.0, taiwan 08 – p.22/41
Ensemble of replicas
4
4
Nrep=25
3
2
xg(x,Q02)
xg(x,Q02)
2
1
1
0
0
-1
-1
-2
1e-05
0.0001
0.001
0.01
0.1
1
-2
1e-05
x
•
•
Nrep=100
3
0.0001
0.001
0.01
0.1
1
x
individual replicas fluctuate significantly
averages are smooth as the number of replicas is increased
nnpdf1.0, taiwan 08 – p.23/41
PDFs uncertainties
•
Monte Carlo prescription (NNPDF)
σF =
•
«1/2
´
Nset `
hF [{f }]2 i − hF [{f }]i2
Nset − 1
„
HEPDATA prescription (CTEQ and MRST/MSTW)
σF
0
Nset /2 “
1 @ X
=
2C90
k=1
F [{f
(2k−1)
}] − F [{f
(2k)
}]
”2
11/2
A
,
C90 = 1.64485
C90 accounts for the fact that the upper and lower parton sets correspond to 90%
confidence levels rather than to one-σ uncertainties.
•
HEPDATA∗ prescription (Alekhin)
0
N
set
X
σF = @
k=1
11/2
“
”2
(k)
(0)
F [{f }] − F [{f }] A
.
nnpdf1.0, taiwan 08 – p.24/41
The NNPDF1.0 parton set
6
4
CTEQ6.5
MRST2001E
CTEQ6.5
MRST2001E
Alekhin02
5
Alekhin02
3
NNPDF1.0
NNPDF1.0
2
xg (x, Q2)
0
0
x Σ (x, Q 2)
4
3
1
2
0
1
-1
0 -5
10
-3
10-4
10
x
10-2
10-1
-2 -5
10
1
CTEQ6.5
MRST2001E
10
10
x
10-2
10-1
1
CTEQ6.5
MRST2001E
10
Alekhin02
Alekhin02
NNPDF1.0
NNPDF1.0
1
0
0
g (x, Q2)
1
Σ (x, Q 2)
-3
10-4
10-1
10-2
10-2
-3
10
10-1
-3
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
10
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
nnpdf1.0, taiwan 08 – p.25/41
The NNPDF1.0 parton set
60
40
CTEQ6.5
MRST2001E
Alekhin02
50
Alekhin02
30
NNPDF1.0
NNPDF1.0
25
0
0
T3 (x, Q 2)
40
V (x, Q2)
CTEQ6.5
MRST2001E
35
30
20
20
15
10
5
0
10
-5
0 -3
10
10-2
-10 -3
10
10-1
x
10-1
x
0.1
1.4
CTEQ6.5
MRST2001E
1.2
CTEQ6.5
MRST2001E
0.08
Alekhin02
Alekhin02
NNPDF1.0
1
NNPDF1.0
0.06
0
x ∆ S (x, Q 2 )
0.8
0.04
0
xV (x, Q2)
10-2
0.6
0.02
0.4
0
0.2
-0.02
0
-0.2
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
-0.04
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
nnpdf1.0, taiwan 08 – p.26/41
Relative uncertainties
0.3
0.6
CTEQ6.5
MRST2001E
CTEQ6.5
MRST2001E
Alekhin02
0.2
Alekhin02
0.4
-0.1
-0.2
-0.3 -5
10
•
•
0
0.2
0
0
NNPDF1.0
σ( g(x,Q 2) )/g(x,Q2 )
0
0.1
0
σ( Σ(x,Q2) )/ Σ(x,Q2)
NNPDF1.0
0
-0.2
-0.4
10-4
-3
10
x
10-2
10-1
-0.6
10-4
-3
10
10-2
x
10-1
comparable uncertainty in the data region
BUT larger uncertainties in the extrapolation
nnpdf1.0, taiwan 08 – p.27/41
PDFs correlations
Correlations between u − u and g − g (Q=85GeV)
ρ
ˆ
fa (x1 , Q21 )fb (x2 , Q22 )
˜
[nadolsky 08]
hfa (x1 , Q21 )fb (x2 , Q22 )irep − hfa (x1 , Q21 )irep hfb (x2 , Q22 )irep
=
.
σa (x1 , Q21 )σb (x2 , Q22 )
nnpdf1.0, taiwan 08 – p.28/41
Distance between MC ensembles.
•
•
Stability of the NNPDF parton set can be assessed by using standard statistical
tools.
n
o
(1)
(1)
2
Distances between two probability distributions: fik = fk (xi , Q0 )
v “
”2 +
u*
u
hfi i(1) − hfi i(2)
u
t
hd[f ]i =
(1)
(2)
σ 2 [fi ] + σ 2 [fi ] pts
•
where:
(1)
hfi i(1) ≡
1
(1)
Nrep
Nrep
X
(1)
fik ,
k=1
(1)
σ
•
2
(1)
[fi ]
≡
1
(1)
(1)
Nrep (Nrep − 1)
Nrep
”2
X “ (1)
fik − hfi i(1)
k=1
For statistically equivalent PDF sets: hd[f ]i ∼ hd[σf ]i ∼ 1
nnpdf1.0, taiwan 08 – p.29/41
Stability under variation of the parametrization
Data
Extrapolation
Σ(x, Q2
0)
hd[f ]i
5 10−4 ≤ x ≤ 0.1
10−5 ≤ x ≤ 10−4
0.98
1.25
2
hd[σ]i
1.14
1.34
1
g(x, Q2
0)
5 10−4 ≤ x ≤ 0.1
10−5 ≤ x ≤ 10−4
hd[f ]i
1.52
1.15
hd[σ]i
1.16
1.07
0.05 ≤ x ≤ 0.75
10−3 ≤ x ≤ 10−2
1.00
1.11
hd[σ]i
1.76
2.27
V (x, Q2
0)
0.1 ≤ x ≤ 0.6
3 10−3 ≤ x ≤ 3 10−2
hd[f ]i
1.30
0.90
hd[σ]i
1.10
0.98
∆S (x, Q2
0)
0.1 ≤ x ≤ 0.6
3 10−3 ≤ x ≤ 3 10−2
hd[f ]i
1.04
1.91
hd[σ]i
1.44
1.80
4
CTEQ6.5
MRST2001E
Alekhin02
3
0
xg (x, Q2)
NNPDF1.0
0
T3 (x, Q2
0)
hd[f ]i
-1
-2 -5
10
10-4
-3
10
x
10-2
10-1
1
Gluon PDF
4
CTEQ6.5
MRST2001E
Alekhin02
3
NNPDF08 prelim.
0
xg (x, Q2)
2
1
•
0
-1
-2 -5
10
Stability under change of architecture of the
nets:
37 pars → 31 pars
10-4
10
-3
x
10-2
10-1
1
•
Independence on the parametrization!
nnpdf1.0, taiwan 08 – p.30/41
Dependence on data sets
HERA-LHC benchmark
Benchmark PDF fit to a reduced consistent set of DIS data
104
[hep-ph/0511119]
NNPDF1.0
NNPDF_bench_H-L
3
Q 2 (GeV2)
10
102
10
1 -5
10
-3
10-4
10
3163
x
10-2
10-1
1
data −→ 773 data
nnpdf1.0, taiwan 08 – p.31/41
Dependence on data sets
HERA-LHC benchmark
Comparison between collaborations and between benchmark/global partons.
u(x, Q2 = 2GeV2 ): MRST data region
1.2
NNPDF1.0
1
NNPDF1.0 [bench*]
MRST2001E
MRST bench
0
xu(x,Q2)
0.8
0.6
0.4
0.2
0
10-2
10-1
x
1
nnpdf1.0, taiwan 08 – p.32/41
Dependence on data sets
HERA-LHC benchmark
Comparison between collaborations and between benchmark/global partons.
u(x, Q2 = 2GeV2 ): MRST extrapolation region
1.2
NNPDF1.0
NNPDF1.0 [bench*]
1
MRST2001E
MRST bench
0
xu(x,Q2)
0.8
0.6
0.4
0.2
0
-5
10
10-4
-3
10
x
10-2
10-1
1
nnpdf1.0, taiwan 08 – p.33/41
Dependence on data sets
HERA-LHC benchmark
•
•
benchmark partons and global partons do not agree within error!
•
not satisfactory especially to predict the behaviour of PDFs in the extrapolation
region (LHC)
note that PDFs input parametrization, flavor assumptions and statistical treatment
(∆χ2global = 50, ∆χ2bench = 1) are tuned to data.
nnpdf1.0, taiwan 08 – p.34/41
Dependence on data sets
HERA-LHC benchmark
Comparison between collaborations and between benchmark/global partons.
u(x, Q2 = 2GeV2 ): data region
1.2
NNPDF1.0
1
NNPDF1.0 [bench*]
MRST2001E
MRST bench
0
xu(x,Q2)
0.8
0.6
0.4
0.2
0
10-2
10-1
x
1
nnpdf1.0, taiwan 08 – p.35/41
Dependence on data sets
HERA-LHC benchmark
Comparison between collaborations and between benchmark/global partons.
u(x, Q2 = 2GeV2 ): extrapolation region
1.2
NNPDF1.0
NNPDF1.0 [bench*]
1
MRST2001E
MRST bench
0
xu(x,Q2)
0.8
0.6
0.4
0.2
0
-5
10
10-4
-3
10
x
10-2
10-1
1
nnpdf1.0, taiwan 08 – p.36/41
Dependence on data sets
HERA-LHC benchmark
•
•
•
•
•
NNPDF1.0 is consistent with MRST global fit.
NNPDFbench is consistent with NNPDF1.0 and MRST.
Same parametrization and flavour assumption.
Same statistical treatment.
Underestimation of the error in the standard approach.
nnpdf1.0, taiwan 08 – p.37/41
Comparison with present experimental data
0.7
2.2
CTEQ6.5
CTEQ6.5
MRST2001E
Alekhin02
0.6
Alekhin02
0.5
0.4
2
2
(x, y, Q2=12 GeV2 )
NNPDF1.0
F2 (x, Q =15 GeV )
MRST2001E
2
νp
p
0.3
σ
0.2
NNPDF1.0
1.8
1.6
1.4
1.2
1
0.8
0.1
0.6
0
-2
-1
10
x
10
0.1
1
0.15
0.2
0.25
0.3
x
0.35
0.4
0.45
0.5
0.6
CTEQ6.5
CTEQ6.5
MRST2001E
1.4
MRST2001E
1
0.8
e+ p
0.6
NNPDF1.0
0.4
0.3
0.2
0.1
0.4
10-4
Alekhin02
0.5
σCC (x, y, Q2=2000 GeV2 )
NNPDF1.0
1.2
e +p
σNC (x, y, Q2=6.5 GeV2 )
Alekhin02
-3
10
x
10-2
0.05
0.1
0.15
x
0.2
0.25
nnpdf1.0, taiwan 08 – p.38/41
LHC standard candle processes
•
•
All quantities have been computed at NLO with MCFM
[http://mcfm.fnal.gov]
Quoted uncertainties are the 1σ bands due to the PDF uncertainty only.
σW + Bl+ ν
∆σW + /σW +
σZ Bl+ l−
∆σZ /σZ
NNPDF1.0
11.83 ± 0.26
2.2%
1.95 ± 0.04
2.1%
CTEQ6.1
11.65 ± 0.34
2.9%
1.93 ± 0.06
3.1%
MRST01
11.71 ± 0.14
1.2%
1.97 ± 0.02
1.0%
CTEQ6.5
12.54 ± 0.29
2.3%
2.07 ± 0.04
1.9%
l
W+ Cross Section at the LHC [MCFM]
Z0 Cross Section at the LHC [MCFM]
12.8
2.1
12.6
2.05
12.2
σ(Z0)Bl+l- [nb]
σ(W+)Blν [nb]
12.4
12
11.8
11.6
2
1.95
1.9
11.4
1.85
11.2
NNPDF1.0
11
CTEQ6.1
MRST2001E CTEQ6.5
NNPDF1.0
CTEQ6.1
MRST2001E CTEQ6.5
1.8
nnpdf1.0, taiwan 08 – p.39/41
Conclusions
•
Standard approaches with fixed parametrization tend to underestimate
uncertainties unless experimental errors are inflated by essentially arbitrary
amount.
•
Monte Carlo ensemble
•
•
◦
Any statistical property of PDFs can be calculated using standard statistical
methods.
◦
No need of any tolerance criterion.
The Neural Network parametrization
◦
Small uncertainties come from an underlying physical law, not from
parametrization bias.
◦
Inconsistent data or underestimated uncertainties do not require a separate
treatment and are automatically signalled by a larger value of the χ2 .
The first NNPDF parton set [arXiv:0808.1231] is available on the common LHAPDF
interface [http://projects.hepforge.org/lhapdf].
nnpdf1.0, taiwan 08 – p.40/41
Outlook
•
Inclusion of hadronic data to
◦
◦
◦
improve the accuracy of gluon at large x (jets)
determine the light antiquark sea asymmetry (Drell-Yan)
allow for a direct determination of the strange distribution (dimuon data)
•
More accurate treatment of Heavy Quark thresholds.
•
LO parton set in view of its use in Monte Carlo generators.
•
More sophisticated theoretical treatment: NNLO parton distributions, large and
small x resummation corrections should also be considered.
•
Study of the impact of PDFs uncertainties on LHC phenomenology
nnpdf1.0, taiwan 08 – p.41/41