parton distribution functions and their collider applications
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Parton distribution functionsand
their collider applications
Pavel Nadolsky (SMU)J. Huston, H.-L. Lai, J.Pumplin, D. Stump, W.-K. Tung, C.-P. Yuan
progress on CT09 PDF set
Rik Yoshida and Steve McGill (ANL)HERAPDF0.2 set
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 1
Wu-Ki Tung1939-2009
¥ Professor of Physics at IIT, Michigan State Universityand University of Washington; well known for his work onhadronic physics
¥ A founder and long-term leader of the CoordinatedTheoretical-Experimental Project on QCD (CTEQ)
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 2
The 1990 paper by Morfinand Tung pioneered theglobal QCD analysis ofparton distributionfunctions (in parallel withthe effort by Martin,Roberts, and Stirling inEurope)
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 3
Many common themesof ongoing PDF studies(interplay of constraintsfrom differentexperiments, PDFuncertainties, predictionsfor “precisionobservables”, ...) arealready present in the M-Tpaper
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 4
Many common themesof ongoing PDF studies(interplay of constraintsfrom differentexperiments, PDFuncertainties, predictionsfor “precisionobservables”, ...) arealready present in the M-Tpaper
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 4
Parton distribution functions in 2009
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 5
Parton distribution functions fa/p(x,Q)...¥ ...are universal nonperturbative functions needed for manyperturbative QCD calculations
¥ ... are parametrized as
fi/p(x,Q0) = a0xa1(1− x)a2F (a3, a4, ...) at Q0 ∼ 1 GeV
¥ ... are found from Dokshitzer-Gribov-Lipatov-Altarelli-Parisi
(DGLAP) equations at Q > Q0:
Qdfi/p(x,Q)
dQ=
∑
j=g,u,u,d,d,....
∫ 1
x
dy
yPi/j
(x
y, αs(Q)
)fj/p(y, Q),
with Pi/j known to order α3s (NNLO):
Pi/j (x, αs) = αsP(1)i/j (x) + α2
sP(2)i/j (x) + α3
sP(3)i/j (x) + ...
¥ Free parameters ai and their uncertainties are determined
from a global fit to hadron scattering dataPavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 6
Parton distributions for the Large Hadron ColliderPDF’s must be determined in awide (x,Q) range with accuracy∼ 1% for purposes of...
¥ monitoring of the LHCluminosity, calibration ofdetectors
¥ tests of electroweak symmetrybreaking (EWSB)
¥ searches for Higgs bosons,supersymmetry, etc
¥ discrimination between newphysics models
¥ precision tests of hadronicstructure
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
100
101
102
103
104
105
106
107
108
109
fixedtarget
HERA
x1,2
= (M/14 TeV) exp(±y)Q = M
LHC parton kinematics
M = 10 GeV
M = 100 GeV
M = 1 TeV
M = 10 TeV
66y = 40 224
Q2
(GeV
2 )
x
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 7
Key Tevatron/LHC measurements require trustworthy PDFs
For example, leading syst. uncertainties in tests of electroweaksymmetry breaking are due to insufficiently known PDFs
EW precision fits
0
1
2
3
4
5
6
10030 300
mH [GeV]
∆χ2
Excluded Preliminary
∆αhad =∆α(5)
0.02758±0.00035
0.02749±0.00012
incl. low Q2 data
Theory uncertaintyJuly 2008 mLimit = 154 GeV
A large part of δMH arises from δP DF MW
EW fits + direct Higgs searches
160 165 170 175 180 185mt [GeV]
80.20
80.30
80.40
80.50
80.60
80.70
MW
[GeV
]SM
MSSM
MH = 114 GeV
MH = 400 GeV
light SUSY
heavy SUSY
SMMSSM
both models
Heinemeyer, Hollik, Stockinger, Weber, Weiglein ’07
experimental errors 68% CL:
LEP2/Tevatron (today)
Tevatron/LHC
SM band: 114 ≤ MH ≤ 400 GeVSUSY band: random scan
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 8
Origin of differences between PDF sets
1. Corrections of wrong or outdated assumptions
lead to significant differences between new (≈post-2007) andold (≈pre-2007) PDF sets
¥ inclusion of (N)NLO QCD, heavy-quark hard scatteringcontributions
I CTEQ6.6 and MSTW’2008 PDFs implement completeheavy-quark treatment; previous PDFs are obsolete without it
I “NNLO” contributions are not automatically equivalent tobetter theory; to claim that, instabilities at small x or nearheavy-quark thresholds must be also “tamed”
¥ relaxation of ad hoc constraints on PDF parametrizations
¥ improved numerical approximations
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 9
Origin of differences between PDF sets
2. PDF uncertainty
a range of allowed PDF shapes for plausible input assumptions,partly reflected by the PDF error band
is associated with
¥ the choice of fitted experiments
¥ experimental errors propagated into PDF’s
¥ handling of inconsistencies between experiments
¥ choice of factorization scales, parametrizations for PDF’s,higher-twist terms, nuclear effects,...
leads to non-negligible differences between the newest PDF sets
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 9
Nucleon PDFs: selection of experimental data
DIS-based analyses ⇒ focus on the most precise (HERA DIS) data
¥ NC DIS, CC DIS, NC DIS jet, c and b production (H1, ZEUS, HERAPDF)
¥ some fixed-target DIS and Drell-Yan data, compatible with HERADIS at ∆χ2 = 1 level (S. Alekhin)
Global analyses (CT09, MSTW’2008, NNPDF1.1)
⇒ focus on completeness, reliable flavor decomposition
¥ all HERA data + fixed-target DIS data
I notably, CCFR and NuTeV νN DIS constraining s(x,Q)
¥ low-Q Drell-Yan (E605, E866), Run-1 W lepton asymmetry,Run-2 Z rapidity (CT09, MSTW’08, upcoming NNPDF2.0)
¥ Tevatron Run-2 jet production, W asymmetry (CT09, MSTW’08)
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 10
Confidence intervals in global PDF analyses
CTEQ6 tolerance criterion (2001)
¥ acceptable values of PDF parameters must agree at ≈90% c.l.with all experiments included in the fit, for a plausible range oftheoretical assumptions
¥ is realized by accepting all PDF fits with ∆χ2 < T 2 ≈ 100
¥ this criterion is modified in the new CT09 fit (Pumplin et al., arXiv:0904.2424)
-30
-20
-10
0
10
20
30
40
dist
ance
Eigenvector 4
BC
DM
Sp
BC
DM
Sd
H1a
H1b
ZE
US
NM
Cp
NM
Cr
CC
FR2
CC
FR3
E60
5
CD
Fw
E86
6
D0j
et
CD
Fjet
-30
-20
-10
0
10
20
30
40
dist
ance
Eigenvector 18
BC
DM
Sp
BC
DM
Sd
H1a
H1b
ZE
US
NM
Cp
NM
Cr
CC
FR2
CC
FR3
E60
5
CD
Fw
E86
6
D0j
et
CD
Fjet
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 11
Confidence intervals in global PDF analyses
MSTW tolerance criterion (2008)
¥ an evolved version of the original tolerance criterion
¥ T 2 is calculated independently for each PDF eigenvector
¥ is close on average to T 2 ≈ 50 (but for which assumptions?)
Eigenvector number
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
glo
bal
2 χ∆T
ole
ran
ce T
=
-20
-15
-10
-5
0
5
10
15
20
(MRST)50+
(MRST)50-
(CTEQ)100+
(CTEQ)100-
NC
rσ
H1
ep 9
7-00
N
Cr
σH
1 ep
97-
00
2d
Fµ
NM
C
Xµ
µ→
Nν
Nu
TeV
Xµµ
→Nν
Nu
TeV
Xµ
µ→
NνC
CF
R
E86
6/N
uS
ea p
d/p
p D
YE
866/
Nu
Sea
pd
/pp
DY
Xµµ
→Nν
Nu
TeV
3
N x
Fν
Nu
TeV
Xµµ
→N
νN
uT
eV
Xµ
µ→
Nν
Nu
TeV
asy
m.
νl→
II W
∅D
2d
Fµ
BC
DM
S
2d
Fµ
BC
DM
S
2p
Fµ
BC
DM
S
NC
rσ
H1
ep 9
7-00
N
Crσ
ZE
US
ep
95-
00
2d
Fµ
BC
DM
S
2S
LA
C e
d F
NC
rσH
1 ep
97-
00
NC
rσ
ZE
US
ep
95-
00
E86
6/N
uS
ea p
d/p
p D
YE
866/
Nu
Sea
pd
/pp
DY
E86
6/N
uS
ea p
p D
Y3
N x
Fν
Nu
TeV
2d
Fµ
NM
C
asy
m.
νl→
II W
∅D
NC
rσ
H1
ep 9
7-00
2
N F
νN
uT
eV
Xµµ
→N
νC
CF
R
E86
6/N
uS
ea p
d/p
p D
Y
Xµµ
→Nν
Nu
TeV
Xµ
µ→
NνC
CF
R
asy
m.
νl→
II W
∅D
E86
6/N
uS
ea p
d/p
p D
Y
NC
rσ
H1
ep 9
7-00
N
Crσ
H1
ep 9
7-00
3N
xF
νN
uT
eV
Xµµ
→Nν
Nu
TeV
MSTW 2008 NLO PDF fit
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 12
Confidence intervals in global PDF analyses
Neural Network PDF
PDFsUnweightedai
χ2
AveragePDFA very general approach that
¥ realizes stochastic sampling ofthe probability distribution inPDF parameter space(Alekhin; Giele, Keller, Kosower)
¥ parametrizes PDF’s by flexibleneural networks
¥ does not rely on smoothnessof χ2 or Gaussianapproximations
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 13
Effect on W/Z ratio at the LHC
A. Cooper-Sarkar, Workshop on early LHC data, London, March 2009
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 22
Effect on W/Z ratio at the LHC
A. Cooper-Sarkar, Workshop on early LHC data, London, March 2009
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 23
HERAPDF vs. the other PDF sets
¥ The H1+ZEUS sample has a much smaller systematicaluncertainty than the H1 and ZEUS samples individually
¥ Nominally, very small uncertainty compared toCTEQ-MSTW-NNPDF!
¥ However:
I insufficient PDF flavor separation [neutral-current DIS probesonly 4/9 (u + u + c + c) + 1/9
(d + d + s + s
)]
I very rigid PDF parametrizations ⇒ less flexibility to probe allallowed PDF behavior, notably at small x
I typical gluon forms, e.g., g(x,Q0) = AxB(1− x)C (1 + Dx), areruled out by the Tevatron jet data (Pumplin et al., arXiv:0904.2424)
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 25
HERAPDF vs. the other PDF sets
¥ The H1+ZEUS sample has a much smaller systematicaluncertainty than the H1 and ZEUS samples individually
¥ Nominally, very small uncertainty compared toCTEQ-MSTW-NNPDF!
¥ However:
I insufficient PDF flavor separation [neutral-current DIS probesonly 4/9 (u + u + c + c) + 1/9
(d + d + s + s
)]
I very rigid PDF parametrizations ⇒ less flexibility to probe allallowed PDF behavior, notably at small x
I typical gluon forms, e.g., g(x,Q0) = AxB(1− x)C (1 + Dx), areruled out by the Tevatron jet data (Pumplin et al., arXiv:0904.2424)
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 25
NNPDF1.1 vs. other PDFs at Q =√
2 GeV (arXiv:0811.2288)
x-510 -410 -310 -210 -110 1
) 02xg
(x, Q
-2
-1
0
1
2
3
4CTEQ6.6
MRST2001ENNPDF1.0
NNPDF1.1
x-410 -310 -210 -110 1
) 02 (x
, Q+
xs
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2CTEQ6.6
MRST2001ENNPDF1.0
NNPDF1.1
x-510 -410 -310 -210 -110 1
) 02 (x
, QΣx
0
1
2
3
4
5
6CTEQ6.6
MRST2001ENNPDF1.0
NNPDF1.1
At x . 10−3, gluon g, strangenesss+ = (s + s) /2, and singletΣ =
∑i (qi + qi) PDFs are poorly
constrained;
determined by a “theoreticallymotivated” functional form inCTEQ/MSTW, flexible neural net inNNPDF; g, s+ can be < 0!
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 26
Strangeness and σZ/σW at the LHC
10-5 10-4 10-3 0.01 0.02 0.05 0.1 0.2 0.5 0.7x
-1
-0.5
0
0.5
1
Cor
rela
tion
Correlation between ΣZ�ΣW± HLHCL and fHx,Q=85. GeVL
d-
u-
gudscb
The PDF uncertainty inσZ/σW is mostly driven bys(x,Q); increases by a factorof 3 compared to CTEQ6.1as a result of freestrangeness in CTEQ6.6
a stumbling block in the precision measurement of W bosonmass MW at the LHC
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 27
Correlation analysis for collider observables(J. Pumplin et al., PRD 65, 014013 (2002); P.N. and Z. Sullivan, hep-ph/0110378)
A technique based on the Hessian method
For 2N PDF eigensets and two cross sections X and Y :
∆X =1
2
√√√√N∑
i=1
(X
(+)i −X
(−)i
)2
cosϕ =1
4∆X ∆Y
N∑
i=1
(X
(+)i −X
(−)i
) (Y
(+)i − Y
(−)i
)
X(±)i are maximal (minimal) values of Xi tolerated along the i-th PDF
eigenvector direction; N = 22 for the CTEQ6.6 set
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 28
Correlation angle ϕ
Determines the parametric form of the X − Y correlation ellipse
X = X0 + ∆X cos θ
Y = Y0 + ∆Y cos(θ + ϕ)
δX
δY
δX
δY
δX
δY
cos ϕ ≈ 1 cos ϕ ≈ 0 cos ϕ ≈ −1
X0, Y 0: best-fitvalues
∆X, ∆Y : PDF errors
cosϕ ≈ ±1 :cosϕ ≈ 0 :
Measurement of X imposestightloose
constraints on Y
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 29
Strangeness and σZ/σW at the LHC
10-5 10-4 10-3 0.01 0.02 0.05 0.1 0.2 0.5 0.7x
-1
-0.5
0
0.5
1
Cor
rela
tion
Correlation between ΣZ�ΣW± HLHCL and fHx,Q=85. GeVL
d-
u-
gudscb
The PDF uncertainty inσZ/σW is mostly driven bys(x,Q); increases by a factorof 3 compared to CTEQ6.1as a result of freestrangeness in CTEQ6.6
a stumbling block in the precision measurement of W bosonmass MW at the LHC
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 30
cos ϕ for various NLO Higgs productioncross sections in SM and MSSM
Particle mass (GeV)100 150 200 250 300 350 400 450 500
Cor
rela
tion
cosi
ne
−1
−0.5
0
0.5
1sc + bc → h+LHC X-se tion:Correlation with pp → ZX (solid), pp → tt (dashes), pp → ZX (dots)
gg → h0 bb → h0 W+h0 h0 via WW fusiont- hannel single top:Z
tt : Z
W+ : W− : Z
Z (Tev-2): Z (LHC)t- hannel single top: tt
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 31
Correlations between dσ(pp → Z0X)/dy and PDF’scosϕ between dσ(pp → Z0X)/dy at the LHC (
√s = 10 TeV)
and PDFs f(x,Q = 85 GeV)
y=0.05
d-
u-
gudsc
10-5 10-4 10-3 10-2 10-1 1-1.0
-0.5
0.0
0.5
1.0
x
Cor
rela
tion:
cosHΦL
y=1.05
d-
u-
gudsc
10-5 10-4 10-3 10-2 10-1 1-1.0
-0.5
0.0
0.5
1.0
x
Cor
rela
tion:
cosHΦL
y=2.05
d-
u-
gudsc
10-5 10-4 10-3 10-2 10-1 1-1.0
-0.5
0.0
0.5
1.0
x
Cor
rela
tion:
cosHΦL
y=3.85
d-
u-
gudsc
10-5 10-4 10-3 10-2 10-1 1-1.0
-0.5
0.0
0.5
1.0
x
Cor
rela
tion:
cosHΦL
Notice thechange insensitivity toparton flavors andthe shift in themost relevant xrange
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 32
Toward CT09 PDF analysis
¥ An update of CTEQ6.6 study (PRD 78, 013004 (2008))
¥ New experimental data in the fit
I CDF Run-2 and D0 Run-2 inclusive jet production
♦ preliminarily explored in J. Pumplin et al., arXiv:0904.0424; P.N., in preparation
I CDF Run-2 lepton asymmetry
I CDF Z rapidity distribution
I low-Q Drell-Yan pT (E288, E605, R209) and Tevatron Run-1,Run-2 Z pT distributions
¥ updated procedure for PDF error estimates
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 33
Inclusive jet production in Tevatron Run-2
0.5
1.0
1.5
|y| < 0.4
DØ Run II-1L = 0.70 fb
= 0.7coneR
50 100 200 3000.0
0.5
1.0
1.5
1.2 < |y| < 1.6
NLO scale uncertainty
0.4 < |y| < 0.8
T = p
Fµ =
RµNLO pQCD
+non-perturbative corrections
50 100 200 300
1.6 < |y| < 2.0
CTEQ6.5M with uncertainties
MRST2004
0.8 < |y| < 1.2
DataSystematic uncertainty
50 100 200 300
2.0 < |y| < 2.4
(GeV)T
p
data
/ th
eory
(GeV)T
p
data
/ th
eory
(GeV)T
p
data
/ th
eory
(GeV)T
p
data
/ th
eory
(GeV)T
p
data
/ th
eory
(GeV)T
p
data
/ th
eory
D0 Coll., arXiv:0802.2400(700 pb−1); CDF results (1.13 fb−1)
¥ (Almost)negligiblestatistical error
¥ MidCone/kT algorithm samples, corrected to parton level
¥ D0 paper:
I “There is a tendency for the data to be lower than the centralCTEQ prediction...”
I “...but they lie mostly within the CTEQ uncertainty band”
¥ non-negligible effect on the CTEQ gluon PDF?
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 34
Impact of Run-2 jet data on CT09 fit
¥ CT09 fit includes all four Run-1 and Run-2 jet data samples
¥ Excellent quality of the fit: χ2 = 2756 for 2898 data points
(Shifted CDF-Run 2data)/CT09
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 35
Impact of Run-2 jet data on CT09 fit
¥ CT09 fit includes all four Run-1 and Run-2 jet data samples
¥ Excellent quality of the fit: χ2 = 2756 for 2898 data points
(Shifted D0-Run 2data)/CT09
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 35
CT09 and CTEQ6.6 are generally compatible
Gluon PDF: CT09 (red), CT66 (blue)
CT09 PDF uncertainty is about the same as CT66 (compensationbetween the Run-2 jet constraints and more flexible g(x, µ))
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 36
CT09 gluon vs. CT66 and MSTW’08 NLO
10-5 10-4 10-3 0.01 0.02 0.05 0.1 0.2 0.5 0.7x
0.6
0.8
1
1.2
1.4
Ratio
toCT
EQ6.
6
g at Q=2. GeVPRELIMINARY
10-5 10-4 10-3 0.01 0.02 0.05 0.1 0.2 0.5 0.7x
0.6
0.8
1
1.2
1.4
Ratio
toCT
EQ6.
6
g at Q=2. GeVPRELIMINARY
Blue: CTEQ6.6Green: CT09Red: MSTW’08 NLO
10-5 10-4 10-3 0.01 0.02 0.05 0.1 0.2 0.5 0.7x
0.6
0.8
1
1.2
1.4
Ratio
toCT
EQ6.
6
g at Q=85. GeVPRELIMINARY
10-5 10-4 10-3 0.01 0.02 0.05 0.1 0.2 0.5 0.7x
0.6
0.8
1
1.2
1.4
Ratio
toCT
EQ6.
6
g at Q=85. GeVPRELIMINARY
Blue: CTEQ6.6Green: CT09Red: MSTW’08 NLO
⇑
Not really supported by CT09analysis
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 37
Constraints of Run-2 data on CT09 PDFs
Black band: g(x, µ)from CT09 fit
Red band: The samefit without Run-2 jetdata
This band is wider thanCT66 because ofadditional freeparameters
Run-2 data impose tangible constraintson the allowed range of g(x, µ)
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 38
Self-consistency of CT09 fit
Are there tensionsin the fit?
1. Are the Run-2 jet data consistent withtheory?
1.1 Are the PDF parametrizations tooflexible/too rigid?
2. Are the new data consistent with otherexperiments?
3. Are the new data consistent with oneanother?
J. Pumplin et al. (arXiv:0904.2424) find that
¥ individual data sets, and data and theory are generallyconsistent with one another
¥ abnormalities exist in the agreement of D0 Run-1 set withother data sets
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 39
Correlated systematic errors (CSE) in jet productionP. Nadolsky, in preparation
CSE for inclusive jets are important. PDF errors areunderestimated without them. CTEQ takes them into accountsince 2000. CSE are provided in two forms:
1. Npt ×Nλ correlation matrix βkα for Nλ random systematicparameters λα
χ2 =∑
e={expt.}
Npt∑
k=1
1
s2k
(Dk − Tk −
Nλ∑
α=1
λαβkα
)2
+
Nλ∑
α=1
λ2α
Dk are Tk are data and theory
sk is the stat.+syst. uncorrelated error
2. Npt ×Npt covariance matrix C = I + ββT
χ2 =∑
e={expt.}(D − T )T
C−1(D − T )
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 40
Comparison of CSE’s for four jet experiments
¥ β (used by CDF Run-1 and 2, D0 Run-2) has several practicaladvantages compared to C (used by D0 Run-1)
¥ Plausibility of β can be checked by the principal componentanalysis (PCA) of β
I Typically only ≈ Nλ/2 combinations of λα (found by PCA) arerelevant for χ2; rank
[ββT
] ≈ Nλ/2 ¿ Npt
¥ C is a large (Npt ×Npt) matrix provided as a “black box”;plausibility of C is harder to verify. C provided by D0 Run-1has irregularities revealed by PCA
I rank [C − I] = rank[ββT
] ≈ Npt = 90 – too large
¥ This suggests that D0 Run-1 CSE’s are overestimated; mayexplain consistently small χ2
D0 Run-1/Npt ∼ 0.3 in fits, otherpeculiarities of D0 Run-1 data observed in the weight scan
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 41
CDF and D0 Run-2 W asymmetry A`(y)
New CDF and D0 Run-2 W lepton asymmetry (in bins of electronpTe and ηe) ; probes u/d in a range of large x values
0.2 0.4 0.6 0.8
-0.5
0.5
1.0 η> 2.6
η ~ 0
Asy
m-u
/d c
orr
elat
ion
x
Correlation of A`(y) in differentηe bins (pTe > 35 GeV) withu(x)/d(x) (H. Schellman)
We find that CDF and D0 A`(y)data disagree in a similarkinematical range (confirming asimilar MSTW finding)
CDF Run-2 A`(y) agrees ok withthe other data
CT09 includes only CDF Run-2A`(y)
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 42
A preliminary fit to CDF and D0 A`(y)
Pull of CDF and D0 data on best-fit W asymmetry
H.-L. Lai, 2009
VERY PRELIMINARY
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 43
Role of heavy flavors in PDF analysesGeneral-mass (GM) scheme(s), currently adopted by CTEQ, MSTW andHERA analyses, strive to provide consistent description of c, b scatteringboth near heavy-quark thresholds and away from them
In 2006 (CTEQ6.5, hep-ph/0611254), it was realized that the GM (and not zero-mass)treatment of c, b mass terms in DIS is essential for predictingprecision W, Z, cross sections at the LHC
Threshold
suppression
x=0.05 µ2=Q
2+4 m
2
F2c
Q2 / GeV
2
χ=x(1+4 m2 / Q
2 )
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
1 10 102
103
NLO 3-flv
LO 3-flv : GF(1)
Naive LO 4-flv : c
(x)
LO
4-f
lv A
CO
T(c)
: c(c)
18.5 19 19.5 20 20.5 21 21.5 22ΣtotHpp®HW±®{ΝLXL HnbL
1.85
1.9
1.95
2
2.05
2.1
2.15
Σto
tHpp®HZ
0®{{-
LXLHn
bL
W± & Z cross sections at the LHC
CTEQ6.6 (GM)
CTEQ6.1 (ZM)
NNLL-NLO ResBos
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 44
Role of heavy flavors in PDF analysesAt NLO, the dominant mass effect is mostly kinematical; propa-gates from DIS into W, Z cross sections through changes in u(x),d(x)
Thorne and Tung (arXiv:0809.0714): can this kinematical effect beapproximately introduced in the widely used ZM scheme, while pre-serving ZM hard cross sections?
Threshold
suppression
x=0.05 µ2=Q
2+4 m
2
F2c
Q2 / GeV
2
χ=x(1+4 m2 / Q
2 )
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
1 10 102
103
NLO 3-flv
LO 3-flv : GF(1)
Naive LO 4-flv : c
(x)
LO
4-f
lv A
CO
T(c)
: c(c)
18.5 19 19.5 20 20.5 21 21.5 22ΣtotHpp®HW±®{ΝLXL HnbL
1.85
1.9
1.95
2
2.05
2.1
2.15
Σto
tHpp®HZ
0®{{-
LXLHn
bL
W± & Z cross sections at the LHC
CTEQ6.6 (GM)
CTEQ6.1 (ZM)
NNLL-NLO ResBos
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 44
Kinematically improved ZM schemes
At NLO, such schemes were indeed developed(P. N., Tung, arXiv:0903:2667)
They depend on a free parameter λ, tuned to approximateeither a ZM DIS cross section or a GM DIS cross section
They can be viewed either
¥ as improved ZM formulations withrealistic c, b kinematics; or
¥ as simplified GM formulations withapproximate ZM hardcross sections
òò
à
à
à
à
à
CTEQ6.6
ZM0
IMΧ
IMc
IMb
Ima
20.0 20.5 21.0 21.5 22.0 22.5 23.01.90
1.95
2.00
2.05
2.10
2.15
2.20
ΣtotHpp ® HW±®{ΝLXL HnbL
Σto
tHpp®HZ
0®{{-LXLHn
bL
NLO W± & Z cross sections at the LHC
We propose to call them “intermediate-mass (IM) schemes”
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 45
Generalized rescaling variable ζ
Realistic GM behavior of PDF’s is reproduced by a generalizedrescaling variable ζ(λ):
x = ζ/(
1 + ζλM2f /Q2
), with 0 ≤ λ . 1
I ζ → 1 as W → M2f
I λ = 0 : ζ ≡ χ = x(1 + M2
f /Q2)
– theACOT-χ variable
I λ & 1: ζ ≈ x (no rescaling)
I ζ ≈ x for Q2 À M2f
1+ �����������Mf
2
Q2 Ζ=ΧACOT HΛ=0L
Ζ=x HΛ®¥L
Λ=0.1
Λ=0.2
Λ=1 Phy
sica
lthr
esho
ld:W=
Mf;Ζ=
1
10-4 0.001 0.01 0.1 1
1
x
Res
calin
gfa
ctor�
xZM fits with λ ≈ 0.15 closely reproduce GM results
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 46
PDF reweighting in Monte-Carlo integration
If X(±)i and ∆X2 =
∑Ni=1
(X
(+)i −X
(−)i
)2/4 are computed in
2N = 44 independent Monte-Carlo runs with N events each,their resulting estimates are given by
X(±)i = X
(±)i + δ
(±)i ∼ X
(±)i +
c
N1/2
and
∆X2
=1
4
N∑
i=1
(X
(+)i −X
(−)i
)2 ∼ ∆X2 +
c′N
N1/2
δ(±)i is a random MC error dependent on the input PDF, arising,
e.g., from importance sampling
As a result of the PDF dependence of δ(±)i , the error ∆X
2 −∆X2 isincreased by a factor N ∼ 22
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 47
PDF reweighting in Monte-Carlo integration
¥ PDF reweighting generates the same sequence of events tocompute each of 2N cross sections
I all δ(±)
i are the same
I ∆X2
= ∆X2
¥ In multi-loop calculations, PDF reweighting saves CPU timedrastically by reducing slow computations of hard-scatteringmatrix elements
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 47
FROOT: a simple interface for Monte-Carlo PDF reweighting
¥ Written in C, can be linked to standalone FORTRAN/C/C++programs
¥ Simple – 170 lines of the code
¥ Writes the output directly into a ROOT ntuple; no need inintermediate PAW ntuples
¥ Flexible; new columns (branches) with PDF weights or eventscan be added into an existing ntuple
¥ Kinematical cuts, selection conditions can be imposed aposteriori in interactive or batch ROOT sessions
¥ implemented in MCFM, ResBos; additional libraries for ROOTanalysis of reweighted ntuples are on the way
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 48
FROOT: a simple interface for Monte-Carlo PDF reweighting
pp_ → (Z0 → e+ e-) X, √S = 1.96 TeV
yZ
σ(Eigenset 1)σ(CTEQ6.6M)
for 900,000 ResBos events
Reweighted
Not reweighted
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04
1.05
-3 -2 -1 0 1 2 3
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 48
Other ongoing work
¥ Combined fit of PDF’s and Drell-Yan pT distributions
¥ PDF’s for leading-order Monte-Carlo programs
¥ consistency and implementation of heavy-flavorcontributions in the global fit at NNLO
¥ constraints on new physics (strongly interactingsuperpartners, etc.)
¥ public C++/ROOT libraries for PDF reweighting for (N)/NLOcalculations and efficient error analysis
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 49
Backup: Rik’s slides
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 50
Backup: Jets
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 65
Comparison of NLO theoretical calculations¥ NLO theoretical uncertainties are at the level 10-20% (D. Soper)
¥ NLO inclusive jet cross sections are currently available from(at least) two groups:
I Ellis-Kunszt-Soper - in CTQ6.6 and our earlier fits
I NLOJet++ (Nagy) + FastNLO (Kluge, Rabbertz, Wobisch)
- in CT09 and MSTW’08
¥ Jon P. explored
I agreement between EKS and FastNLO
I dependence on the choice of scale, jet algorithms, andpartial threshold resummation corrections
¥ The overall agreement/stability at NLO is satisfactory,although not perfect
CT09 uses FastNLO for µ = pT /2 without the thresholdresummation correction
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 66
Comparison of K=NLO/LO from EKS and FastNLO
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 67
Scale dependence of NLO cross section
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 68
χ2 weighting scans
All questions are explored using the χ2 reweighting technique(Collins, Pumplin, hep-ph/0105207)
χ2 =∑
jet expts.wiχ
2i + χ2
non-jet = wχ2jet + χ2
non-jet
wi = 0: experiment i is not included
wi = 1: common choice
wi À wj 6=i: only experiment i matters in the fit
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 69
Self-consistency of CT09 fit
¥ Individual data sets, and data and theory are generallyconsistent with one another
¥ abnormalities in the agreement of D0 Run-1 set with otherdata sets
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 70
Dependence on the gluon PDF parametrizationCT09 uses a more flexibleg(x, µ0) (“par 1”) than CT66
¥ Par 1: g(x, µ0) = A0xA1(1− x)A2
× eA3x+A4x2+A5x
1/2
¥ CT66: par 1 with A2 = 4, A5 = 0
¥ Par 2: A4 = A5 = 0
¥ Par 3 (H1-like):g(x, µ0) = A0x
A1(1− x)A2 (1 + A3x)
CT09 (par1) form provides thebest χ2 and vanishing tensionwith the non-jet data
300 350 400 450 500 550Χ2
jet
0
50
100
150
200
250
300
DΧ
non-
jet2
Weight scan for Run-1 and Run-2 jet data
Par1
Par2
Par3
wjet = 0 (right ends); 1 (bend);10 (left end)
Par 2 and 3 are disfavored
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 71
Lagrange multiplier method vs. Hessian method
The Hessian method(48 error PDFs — redband) underestimatesthe true δPDF g(x,Q)suggested by χ2
(revealed by the LMmethod — individuallines)
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 72
Backup: IM scheme
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 73
Intermediate-mass scheme: a basic recipe¥ Start with PQCD factorization for DIS, in a form applicable both in
ZM or GM schemes
Fλ(x,Q2) =∑
a,b
∫ 1
ζ
dξ
ξfa(ξ, µ)Ca
b,λ
(ζ
ξ,Q
µ,mi
µ, αs(µ)
)
¥ sum over initial-state active flavors a prescribed by thefactorization scheme for the PDFs
¥ sum over final-state quark flavors b physically produced at thegiven scattering energy
¥ evaluate convolutions over the kinematical range ζ ≤ ξ ≤ 1determined by a rescaling variable ζ
¥ use zero-mass Wilson coefficients Cab,λ=Ca
b,λ
(ζξ , Q
µ , 0, αs(µ))
,
evaluated at ζ/ξ
¥ keep µ > mQ in heavy-quark channels for all Q, to guaranteeapplicability of the (subtracted) MS expression for Ca
b,λ;e.g., use µ =
√Q2 + m2
i in qiγ∗ → qi and gγ∗ → qiqi channels
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 74
Rescaling variables in heavy-flavor DISRescaled light-cone variable ζ is a simple way to approximateexact scattering kinematics in processes where the exactmomentum conservation is absent
¥ General-mass scheme: dominant heavy-quark mass effectsare approximated in LO cγ∗ → c (or sW → c) by a rescalingvariable χ :
x = χ/(
1 + M2f /Q2
), with M2
f = 4m2c (m2
c) in NC (CC) DISBarnett, Haber, Soper; Tung, Kretzer, Schmidt
¥ It is natural to try ζ = χ both in cγ∗ → c and gγ∗ → cc in the IMscheme; however, it leads to excessive suppression of charmscattering in NC DIS at small x (for given Q), where thresholdeffects should be less pronounced, while the PDF variation israpid
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 75
Comparison to c, b SIDIS datafrom CT6.6 data sampleZM/GM/IM Wilson coefficients with ZM (CT6.1M) and GM (CT6.6M) PDF’s
¥ CT6.1 are refittedincluding the latest HERAc, b data (χ2 improved)
¥ GM+CT6.6 gives thebest fit; ZM+CT6.6 theworst
¥ χ2 for IM+CT6.6(IM+CT6.1) is better thanZM+CT6.6 (ZM+6.1)
All IMλ+CT6.x fits produce close χ2 – only IMχ is shown
IM formulation works!
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 76
Comparison to the full CT6.6 data sample
¥GM+CT6.6 still gives thebest fit; ZM+(new CT6.1)the worst
¥ Quality of the best IM fit(IMb) approaches that ofGM+CT6.6
¥ Some IM fits(λ = 0.1− 0.2) are clearlyvery good
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 77
PDFs in GM/IMb/ZM/IMχ formulations
10-5 10-4 10-3 0.01 0.02 0.05 0.1 0.2 0.5 0.70.7
0.8
0.9
1.0
1.1
1.2
1.3
x
Rat
ioto
CTE
Q6.
6M
u(x,Q) at Q = 85 GeV
Black: IM fit with ζ= 0.15Green: IM fit with ζ= χRed: ZM fit Blue: CTEQ6.6 PDF uncertainty
10-5 10-4 10-3 0.01 0.02 0.05 0.1 0.2 0.5 0.7
0.6
0.8
1.0
1.2
1.4
x
Ratio
toCT
EQ6.
6M
g at Q=2. GeV
10-5 10-4 10-3 0.01 0.02 0.05 0.1 0.2 0.5 0.7
0.6
0.8
1.0
1.2
1.4
x
Rat
ioto
CT
EQ
6.6M
s at Q=2. GeV
¥ At x < 0.01, ZM u, g are toosmall compared to GM
¥ IMχ: u, g are too large
¥ IMb: u, g are very close to GM
¥ s(x,Q) (constrained by CCDIS) prefers IMχ rather than IMb
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 78
Dependence on the rescaling ζ
in W , Z production at the LHC
¥ IMλ predictions span a widerange between ZM and IMχ
¥ IM predictions withλ = 0.05− 0.3 are compatiblewith CT6.6M at ≈ 90% c.l.
¥ ζ variable can be alsointroduced in the GM scheme;in this case, the standardchoice ζ = χ (CT6.6, or λ = 0)gives the best χ2
¥ GM PDF’s with λ . 0.3 agreewith CT6.6M within the CT6.6uncertainty
òò
à
à
à
à
à
ó
ó
ó
CTEQ6.6
ZM
IMΧ
IMc
IMb
IMa
GMH0.05L
GMH0.15L
GMH0.3L
20.0 20.5 21.0 21.5 22.0 22.5 23.01.90
1.95
2.00
2.05
2.10
2.15
2.20
ΣtotHpp ® HW±®{ΝLXL HnbL
Σto
tHpp®HZ
0®{{-LXLHn
bL
NLO W± & Z cross sections at the LHC
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 79
Conclusions
¥ The full GM heavy-quark kinematical dependence isapproximated well at NLO by an effective IM scheme with ZMWilson coefficients and generalized rescaling variable ζ
I It remains to be seen if this scheme is viable beyond NLO
¥ The IM formulation can be applied to easily implement theleading heavy-quark mass effects in ZM calculations
¥ Variations in the form of ζ lead to an additional theoreticaluncertainty that must be provided with GM predictions
Pavel Nadolsky (SMU) ANL/IIT workshop, Chicago 05/21/09 80
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