Post on 29-Jan-2021
Standard Model Candles, Theory Uncertainties and
PDFs 2
Robert Thorne
January 11th, 2011
University College London
YETI – January 2011
How straightforward is it in practice?
) (nb)ν± l→ ± B(W⋅ ±Wσ
9.4 9.6 9.8 10 10.2 10.4 10.6 10.8 11
) (
nb)
- l+ l→ 0
B(Z
⋅ 0Zσ
0.8
0.85
0.9
0.95
1
= 7 TeV)sNLO W and Z cross sections at the LHC (
SαOuter error bars: PDF+Inner error bars: PDF only
R(W/Z) = 10.810.911.0
68% C.L. PDF
MSTW08
CTEQ6.6
NNPDF2.0
HERAPDF1.0
ABKM09
GJR08
) (nb)ν± l→ ± B(W⋅ ±Wσ
9.4 9.6 9.8 10 10.2 10.4 10.6 10.8 11
) (
nb)
- l+ l→ 0
B(Z
⋅ 0Zσ
0.8
0.85
0.9
0.95
1
Predictions (Watt) for W andZ cross-sections for LHC withcommon NLO QCD and vectorboson width effects, and commonbranching ratios, and at 7TeV.
Comparing all groups getsignificant discrepancies betweenthem even for this benchmarkprocess.
Can understand some of thesystematic differences.
Some difference in W/Z ratio.
W, Z total cross-sections best-case scenario.
YETI – January 2011 1
Sources of Variations/Uncertainty
It is vital to consider theoretical/assumption-dependent uncertainties:
● Methods of determining “best fit” and uncertainties.
● Underlying assumptions in procedure, e.g. parameterisations and data used.
● Treatment of heavy flavours.
● PDF and αS correlations.
Responsible for differences between groups for extraction of fixed-order PDFs.
YETI – January 2011 2
Different PDF sets
● MSTW08 – fit all previous types of data. Most up-to-date Tevatron jet data. Notmost recent HERA combination of data. PDFs at LO, NLO and NNLO.
● CTEQ6.6 – very similar. Not quite as up-to-date on Tevatron data. PDFs at NLO.New – CT10 include HERA combination and more Tevatron data. Little changes.
● NNPDF2.0 – include all except HERA jet data (not strong constraint) and heavyflavour structure functions. Include HERA combined data. PDFs at NLO.
● HERAPDF2.0 – based entirely on HERA inclusive structure functions, neutral andcharged current. Use combined data. PDFs at LO, NLO.
● ABKM09 – fit to DIS and fixed target Drell-Yan data. PDFs at NLO and NNLO.(Now prelim results using Tevatron jets).
● GJR08 – fit to DIS, fixed target Drell-Yan and Tevatron jet data. PDFs at NLOand NNLO.
Use of HERA combined data instead of original data slight increase in quarks at lowx (depending on procedure).
YETI – January 2011 3
Parton Fits and Uncertainties. Two main approaches.
Parton parameterization and Hessian (Error Matrix) approach first used by H1 andZEUS, and extended by CTEQ. Now used in some form by all other than NNPDF.
χ2 − χ2min ≡ ∆χ2 =∑
i,j
Hij(ai − a(0)i )(aj − a(0)j )
The Hessian matrix H is related to the covariance matrix of the parameters by
Cij(a) = ∆χ2(H−1)ij.
We can then use the standard formula for linear error propagation.
(∆F )2 = ∆χ2∑
i,j
∂F
∂ai(H)−1ij
∂F
∂aj,
This is now the most common approach (sometimes Offset method).
Problematic due to extreme variations in ∆χ2 in different directions in parameterspace.
YETI – January 2011 4
Solved by finding and rescaling eigenvectors of H leading to diagonal form
∆χ2 =∑
i
z2i
Implemented by CTEQ, then others. Uncertainty on physical quantity then given by
(∆F )2 =∑
i
(
F (S(+)i ) − F (S
(−)i )
)2,
where S(+)i and S
(−)i are PDF sets displaced along eigenvector direction.
Question of choosing “correct” ∆χ2 given complication of errors in full fit andsometimes conflicting data sets.
CTEQ use ∆χ2 ∼ 40 and MRST/MSTW use more complicated approach – results in∆χ2 ∼ 15, for one σ. Other fits less global, keep to ∆χ2 = 1.
YETI – January 2011 5
● MSTW08 – 20 eigenvectors. Due to incompatibility of different sets and (perhapsto some extent) parameterisation inflexibility (little direct evidence for this) haveinflated ∆χ2 of 5 − 20 for eigenvectors.
● CTEQ6.6 – 22 eigenvectors. Inflated ∆χ2 of 40 for 1 sigma for eigenvectors(no normalization uncertainties in CTEQ6.6). CT10 have 26 eigenvectors, includenormalization uncertainties and have a slightly modified tolerance criterion.
● HERAPDF2.0 – 9 eigenvectors. Use “∆χ2 = 1′′. Additional model andparameterisation uncertainties.
● ABKM09 – 21 parton parameters. Use ∆χ2 = 1. Also αS,mc,mb.
● GJR08 – 20 parton parameters and αS. Use ∆χ2 ≈ 20. Impose strong theory
constraint on input form of PDFs.
Perhaps surprisingly all get rather similar uncertainties for PDFs cross-sections.
YETI – January 2011 6
Neural Network group (Ball et al.) limit parameterization dependence.
First part of approach, no longer perturb about best fit. Construct a set of Monte
Carlo replicas F art,ki,p of the original data set Fexp,(k)i,p .
Where r(k)p are random numbers following Gaussian distribution, and S
(k)p,N is the
analogous normalization shift of the of the replica. Errors represented in variations of
replicas. Fit to the data replicas obtaining PDF replicas q(net)(k)i (follows Giele et al.)
Mean µO and deviation σO of observable O then given by
µO =1
Nrep
Nrep∑
1
O[q(net)(k)i ], σ
2O =
1
Nrep
Nrep∑
1
(O[q(net)(k)i ] − µO)2.
Eliminates parameterisation dependence by using a neural net which undergoes aseries of (mutations via genetic algorithm) to find the best fit. In effect is a muchlarger sets of parameters – ∼ 37 per distribution. Pre-processing exponents as x → 0and x → 1used to aid convergence.
Each fit performed with half data fit (training) and half checked (validation) andstopped when comparison to validation set deteriorates.
YETI – January 2011 7
Also reductions due to inclusion of new data.
NNPDF uncertainties pretty similar to other groups, with some particular exceptions.
YETI – January 2011 8
Gluon Parameterisation - small x – different parameterisations lead to very differentuncertainty for small x gluon.
x-510 -410 -310 -210 -110
Fra
ctio
nal
un
cert
ain
ty
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
2 = 5 GeV2Gluon distribution at Q
MSTW 2008 NLO (90% C.L.)
CTEQ6.6 NLO
Alekhin 2002 NLONNPDF1.0 (1000 replicas)
x-510 -410 -310 -210 -110
Fra
ctio
nal
un
cert
ain
ty
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Most assume single power xλ at input → limited uncertainty. If input at low Q2 λpositive and small-x input gluon fine-tuned to ∼ 0. Artificially small uncertainty.If g(x) ∝ xλ±∆λ then ∆g(x) = ∆λ ln(1/x) ∗ g(x).MRST/MSTW and NNPDF more flexible (can be negative) → rapid expansion ofuncertainty where data runs out.
YETI – January 2011 9
Generally high-x PDFs parameterisedso will behave like (1 − x)η asx → 1. More flexibility in CTEQ.
Very hard high-x gluon distribution(more-so even than NNPDFuncertainties).
However, is gluon, which isradiated from quarks, harder thanthe up valence distribution forx → 1?
YETI – January 2011 10
PDF correlation with αS.
Can also look at PDF changes and uncertainties at different αS(M2Z). Latter usually
only for one fixed αS(M2Z). Can be determined from fit, e.g. αS(M
2Z) = 0.1202
+0.0012−0.0015
at NLO and αS(M2Z) = 0.1171
+0.0014−0.0014 at NNLO from MSTW.
PDF uncertainties reduced since quality of fit already worse than best fit.
x-410 -310 -210 -110
Rat
io t
o M
ST
W 2
008
NN
LO
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04
1.05
2 = (120 GeV)2H = M 2Gluon at Q
at Tevatron y = 0
at LHCy = 0
MSTW 2008 NNLO (68% C.L.)
at +68% C.L. limitS
αFix
at - 68% C.L. limitS
αFix
x-410 -310 -210 -110
Rat
io t
o M
ST
W 2
008
NN
LO
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04
1.05
Expected gluon–αS(M2Z) small–x anti-correlation → high-x correlation from sum rule.
YETI – January 2011 11
NNLO predictions for Higgs (120GeV) production for different allowed αS(M2Z) values
and their uncertainties.
= 120 GeV) with MSTW 2008 NNLO PDFsH
Higgs (M
)2Z
(MSα∆σ−1 /2σ− 0 /2σ+ σ+1σ−1 /2σ− 0 /2σ+ σ+1
(%
)N
NLO
Hσ∆
−6
−4
−2
0
2
4
6 = 1.96 TeVsTevatron,
)2Z
(MSα∆σ−1 /2σ− 0 /2σ+ σ+1σ−1 /2σ− 0 /2σ+ σ+1
(%
)N
NLO
Hσ∆−6
−4
−2
0
2
4
6 = 14 TeVsLHC,
68% C.L. uncertainties
)2H
(M2S
α
gg luminosity
Increases by a factor of 2−3 (up more than down) at LHC. Direct αS(M2Z) dependencemitigated somewhat by anti-correlated small-x gluon (asymmetry feature of minorproblems in fit to HERA data). At Tevatron intrinsic gluon uncertainty dominates.
YETI – January 2011 12
CTEQ have shown that up to Gaussian approx. for uncertainties (and some othercaveats) αS uncertainty accounted for by adding deviation from PDFs with upperand lower αS limits (red) in quadrature with all other PDF eigenvectors (blue), seenbelow.
)Z
(MSα0.116 0.117 0.118 0.119 0.12
H+X
) [p
b]
→(p
pσ
10.2
10.4
10.6
10.8
11
11.2=120 GeV) at 7 TeV
HHiggs production (m
NNPDF advocate distributing PDF replicas according to probability of αS(m2Z) taking
that value based on some assumed central value and uncertainty, i.e.
NαSrep ∝ exp
(
−(αS−α(0)S
)2
2(δα(68)S
)2
)
,
All lead to roughly same results Vicini et al.
YETI – January 2011 13
Heavy Quarks – Essential to treat these correctly. Two distinct regimes:
Near threshold Q2 ∼ m2H massive quarks not partons. Created in final state. Describedusing Fixed Flavour Number Scheme (FFNS).
F (x,Q2) = CFFk (Q2/m2H) ⊗ f
nfk (Q
2)
Does not sum lnn(Q2/m2H) terms, and not calculated for many processes beyond LO.Still occasionally used. Sometimes final state details in this scheme only.
Alternative, at high scales Q2 À m2H heavy quarks like massless partons. Behavelike up, down, strange. Sum ln(Q2/m2H) terms via evolution. Zero Mass VariableFlavour Number Scheme (ZM-VFNS). Normal assumption in calculations. IgnoresO(m2H/Q2) corrections.
F (x,Q2) = CZMV Fj ⊗ fnf+1
j (Q2).
Need a General Mass Variable Flavour Number Scheme (GM-VFNS) interpolatingbetween the two well-defined limits of Q2 ≤ m2H and Q2 À m2H. Used byMRST/MSTW and more recently (as default) by CTEQ, and now also more regularlyby H1,ZEUS.
YETI – January 2011 14
H1 Fb2b_
(x,Q2)
10-2
10-1
1
10
10 102
103
x=0.0002i=5
x=0.0005i=4
x=0.0013i=3
x=0.005i=2
x=0.013i=1
x=0.032i=0
H1
Q2 / GeV2
Fb 2
b_
× 6
i
H1 DataMSTW08 NNLOMSTW08CTEQ6.6
Various definitions possible. Versionsused by MSTW (RT) and CTEQ(ACOT) have converged somewhat.
Various significant differences stillexist as illustrated by comparisonto most recent H1 data on bottomproduction.
YETI – January 2011 15
Importance of using GM-VFNS instead of masslessapproach illustrated byCTEQ6.5.
Can be > 8% error inPDFs. Much more thanscheme uncertainty.
Leads to large change inpredictions using CTEQpartons at LHC of 5−10%.
YETI – January 2011 16
The values of the predicted cross-sections at NLO for Z and a 120 GeV Higgs bosonat the Tevatron and the LHC (latter for 14 TeV) as GM-VFNS altered.
PDF set Tev LHC (14 TeV)σZ (nb) σH(pb) σZ (nb) σH(pb)
MSTW08 7.207 0.7462 59.25 40.69GMvar1 +0.3% −0.5% +1.1% +0.2%GMvar2 +0.7% −1.1% +3.0% +1.5%GMvar3 +0.1% −0.3% +1.1% +0.8%GMvar4 +0.0% −0.1% −0.4% −0.2%GMvar5 −0.1% −0.1% −0.5% −0.3%GMvar6 +0.3% −0.4% +1.6% +0.8%GMvaropt +0.3% −1.5% +2.0% +0.4%ZM-VFNS −0.7% −1.2% −3.0% −3.1%GMvarcc +0.0% −0.1% +0.0% −0.1%
Little more than 1% variation at Tevatron in σZ.
Up to +3% and −0.5% variation in σZ at the LHC. About half as much in σH dueto higher average x sampled.
Most variation in ZM-VFNS.
YETI – January 2011 17
The values of the predicted cross-sections at NNLO.
PDF set Tev LHC (14 TeV)σZ (nb) σH(pb) σZ (nb) σH(pb)
MSTW08 7.448 0.9550 60.93 50.51GMvar1 +0.1% −0.5% +0.1% −0.2%GMvar2 +0.3% −0.8% +0.5% +0.1%GMvar3 +0.4% −0.1% +0.5% +0.7%GMvar4 +0.0% −0.2% +0.1% −0.1%GMvar5 +0.1% −0.3% −0.2% −0.2%GMvar6 +0.1% −0.9% +0.3% −0.2%GMvaropt +0.4% −0.2% +0.6% +0.8%GMvarmod −0.2% −0.4% −1.4% −1.0%GMvarmod′ +0.0% −0.7% +0.0% +0.1%
Maximum variations of order 1% at LHC. High-x gluon leads to 1% on σH at Tevatron.
Much improved stability compared to NLO.
YETI – January 2011 18
Uncertainties due to mc and mb
Add uncertainties in quadrature with PDF parameter and αS combined uncertainty.
LHC,√
s = 7 TeV B`ν · σW B`+`− · σZ σHCentral value 10.47 nb 0.958 nb 15.50 pb
PDF only uncertainty +1.7%−1.6%
+1.7%−1.5%
+1.1%−1.6%
PDF+αS uncertainty+2.5%−1.9%
+2.5%−1.9%
+3.7%−2.9%
PDF+αS+mc,b uncertainty+2.7%−2.2%
+2.9%−2.4%
+3.7%−2.9%
LHC,√
s = 14 TeV B`ν · σW B`+`− · σZ σHCentral value 21.72 nb 2.051 nb 50.51 pb
PDF only uncertainty +1.7%−1.7%
+1.7%−1.6%
+1.0%−1.6%
PDF+αS uncertainty+2.6%−2.2%
+2.6%−2.1%
+3.6%−2.7%
PDF+αS+mc,b uncertainty+3.0%−2.7%
+3.1%−2.8%
+3.7%−2.8%
NNLO predictions for W , Z and Higgs (MH = 120 GeV) total cross sections or 7 TeVLHC and 14 TeV LHC. Similar results in HERAPDF study Cooper-Sarkar.
αS uncertainties more important, particularly for Higgs. Mass uncertainties significant,but least important of three effects, particularly for Higgs.
YETI – January 2011 19
Predictions by various groups - parton luminosities – NLO. Plots by G. Watt.
/ ss
-310 -210 -110
Rat
io t
o M
ST
W 2
008
NL
O (
68%
C.L
.)
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2 = 7 TeV)sgg luminosity at LHC (
tt120 180 240
(GeV)HM
MSTW08
CTEQ6.6NNPDF2.0
HERAPDF1.0
/ ss
-310 -210 -110
Rat
io t
o M
ST
W 2
008
NL
O (
68%
C.L
.)
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
/ ss
-310 -210 -110
Rat
io t
o M
ST
W 2
008
NL
O (
68%
C.L
.)
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2 = 7 TeV)sgg luminosity at LHC (
tt120 180 240
(GeV)HM
MSTW08
CTEQ6.6ABKM09
GJR08
/ ss
-310 -210 -110
Rat
io t
o M
ST
W 2
008
NL
O (
68%
C.L
.)
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
dLijdŝdy
=1
s
1
1 + δij[fi(x1)fj(x2) + fi(x1)fj(x2)] and integrate over y
Cross-section for tt̄ almost identical in PDF terms to 450GeV Higgs.
Also H + tt̄ at√
ŝ/s ∼ 0.1.
Clearly some distinct variation between groups. Much can be understood in terms ofprevious differences in approaches.
YETI – January 2011 20
/ ss
-310 -210 -110
Rat
io t
o M
ST
W 2
008
NL
O (
68%
C.L
.)
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2 = 7 TeV)s) luminosity at LHC (q(qqΣ
W Z
MSTW08
CTEQ6.6NNPDF2.0
HERAPDF1.0
/ ss
-310 -210 -110
Rat
io t
o M
ST
W 2
008
NL
O (
68%
C.L
.)
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
/ ss
-310 -210 -110
Rat
io t
o M
ST
W 2
008
NL
O (
68%
C.L
.)
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2 = 7 TeV)s) luminosity at LHC (q(qqΣ
W Z
MSTW08
CTEQ6.6ABKM09
GJR08
/ ss
-310 -210 -110
Rat
io t
o M
ST
W 2
008
NL
O (
68%
C.L
.)
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Many of the same general features for quark-antiquark luminosity. Some differencesmainly at higher x.
Canonical example W, Z production, but higher ŝ/s relevant for WH or vector bosonfusion.
All plots and more at http://projects.hepforge.org/mstwpdf/pdf4lhc
YETI – January 2011 21
Variations in Cross-Section Predictions – NLO
)2Z
(MSα0.114 0.116 0.118 0.12 0.122 0.124
(p
b)
Hσ
10.5
11
11.5
12
12.5
13
68% C.L. PDFMSTW08
CTEQ6.6
NNPDF2.0HERAPDF1.0
ABKM09GJR08
= 120 GeVH
= 7 TeV) for MsH at the LHC (→NLO gg
SαOuter: PDF+Inner: PDF onlyVertical error bars
)2Z
(MSα0.114 0.116 0.118 0.12 0.122 0.124
(p
b)
Hσ
10.5
11
11.5
12
12.5
13
)2Z
(MSα0.114 0.116 0.118 0.12 0.122 0.124
(p
b)
Hσ
4.4
4.6
4.8
5
5.2
5.4
68% C.L. PDFMSTW08
CTEQ6.6
NNPDF2.0HERAPDF1.0
ABKM09GJR08
= 180 GeVH
= 7 TeV) for MsH at the LHC (→NLO gg
SαOuter: PDF+Inner: PDF onlyVertical error bars
)2Z
(MSα0.114 0.116 0.118 0.12 0.122 0.124
(p
b)
Hσ
4.4
4.6
4.8
5
5.2
5.4
Dotted lines show how central PDF predictions vary with αS(M2Z).
Plots based on PDF4LHC benchmark criteria, but from extensive independent studyby G. Watt.
Clearly much more variation in predictions than uncertainties claimed by individualgroups.
YETI – January 2011 22
)2Z
(MSα0.114 0.116 0.118 0.12 0.122 0.124
(p
b)
Hσ
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
68% C.L. PDFMSTW08
CTEQ6.6
NNPDF2.0HERAPDF1.0
ABKM09GJR08
= 240 GeVH
= 7 TeV) for MsH at the LHC (→NLO gg
SαOuter: PDF+Inner: PDF onlyVertical error bars
)2Z
(MSα0.114 0.116 0.118 0.12 0.122 0.124
(p
b)
Hσ
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
)2Z
(MSα0.114 0.116 0.118 0.12 0.122 0.124
(p
b)
ttσ
120
130
140
150
160
170
180
190
68% C.L. PDFMSTW08
CTEQ6.6
NNPDF2.0HERAPDF1.0
ABKM09GJR08
= 7 TeV)s cross sections at the LHC (tNLO t
SαOuter: PDF+Inner: PDF onlyVertical error bars
)2Z
(MSα0.114 0.116 0.118 0.12 0.122 0.124
(p
b)
ttσ
120
130
140
150
160
170
180
190
Excluding GJR08 amount of difference due to αS(M2Z) variations 3 − 4%.
CTEQ6.6 now heading back towards MSTW08 and NNPDF2.0.
YETI – January 2011 23
)2Z
(MSα0.114 0.116 0.118 0.12 0.122 0.124
) (
nb
)ν± l
→ ± B
(W⋅ ±
Wσ
9
9.2
9.4
9.6
9.8
10
10.2
10.4
10.6
10.8
11
68% C.L. PDFMSTW08
CTEQ6.6
NNPDF2.0HERAPDF1.0
ABKM09GJR08
= 7 TeV)s at the LHC (ν± l→ ±NLO W
SαOuter: PDF+Inner: PDF onlyVertical error bars
)2Z
(MSα0.114 0.116 0.118 0.12 0.122 0.124
) (
nb
)ν± l
→ ± B
(W⋅ ±
Wσ
9
9.2
9.4
9.6
9.8
10
10.2
10.4
10.6
10.8
11
)2Z
(MSα0.114 0.116 0.118 0.12 0.122 0.124
) (
nb
)- l+ l
→ 0 B
(Z⋅ 0
Zσ
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
68% C.L. PDFMSTW08
CTEQ6.6
NNPDF2.0HERAPDF1.0
ABKM09GJR08
= 7 TeV)s at the LHC (-l+ l→ 0NLO Z
SαOuter: PDF+Inner: PDF onlyVertical error bars
)2Z
(MSα0.114 0.116 0.118 0.12 0.122 0.124
) (
nb
)- l+ l
→ 0 B
(Z⋅ 0
Zσ
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
W+ + W− cross-section. αS(M2Z) dependence now more due to PDF variation with
αS(M2Z).
Again variations somewhat bigger than individual uncertainties.
Roughly similar variation for ŝ up to a few times higher.
YETI – January 2011 24
)2Z
(MSα0.114 0.116 0.118 0.12 0.122 0.124
Zσ ⋅- l+ l /
BWσ ⋅ νl
B≡ W
ZR
10.7
10.75
10.8
10.85
10.9
10.95
11
11.05
68% C.L. PDFMSTW08
CTEQ6.6
NNPDF2.0HERAPDF1.0
ABKM09GJR08
= 7 TeV)sNLO W/Z ratio at the LHC (
SαOuter: PDF+Inner: PDF onlyVertical error bars
)2Z
(MSα0.114 0.116 0.118 0.12 0.122 0.124
Zσ ⋅- l+ l /
BWσ ⋅ νl
B≡ W
ZR
10.7
10.75
10.8
10.85
10.9
10.95
11
11.05
)2Z
(MSα0.114 0.116 0.118 0.12 0.122 0.124
-Wσ
/ +Wσ ≡ ±
R
1.4
1.42
1.44
1.46
1.48
1.5
68% C.L. PDFMSTW08
CTEQ6.6
NNPDF2.0HERAPDF1.0
ABKM09GJR08
= 7 TeV)s ratio at the LHC (-/W+NLO W
SαOuter: PDF+Inner: PDF onlyVertical error bars
)2Z
(MSα0.114 0.116 0.118 0.12 0.122 0.124
-Wσ
/ +Wσ ≡ ±
R
1.4
1.42
1.44
1.46
1.48
1.5
For W/Z values consistent but uncertainties vary. Largely due to strange uncertainty.
Quite a variation in ratio for W+/W−. Shows variations in flavour and quark-antiquarkdecompositions.
All plots and more at http://projects.hepforge.org/mstwpdf/pdf4lhc
YETI – January 2011 25
Entries 0Mean 0RMS 0
W+y
-4 -3 -2 -1 0 1 2 3 4
Rat
io t
o C
TE
Q6.
6
0.85
0.9
0.95
1
1.05
1.1
1.15Entries 0Mean 0RMS 0
CTEQ6.6
ABKM09
GJR08
HERAPDF1.0
MSTW2008
NNPDF2.0
Differences also clear inrapidity distributions.
Plot from PDF4LHC InterimReport.
Shape discriminating even ifnormalisation will be difficultfor a while.
YETI – January 2011 26
Translates into some significantdifferences in the more different W -asymmetry predictions.
MSTW08 and CTEQ6.6 about thebiggest discrepancy at low y.
HERAPDF diverges from others athighest y.
Possibly first real discriminating powerfrom LHC measurements.
YETI – January 2011 27
Deviations In predictions clearly much more than uncertainty claimed by each.
In some cases clear reason why central values differ, e.g. lack of some constrainingdata, though uncertainties then do not reflect true uncertainty.
Sometimes no good understanding, or due to difference in procedure which is simplya matter of disagreement, e.g. gluon parameterisation at small x affects predictedHiggs cross-section.
What is true uncertainty. Task asked of PDF4LHC group.
Interim recommendation take envelope of global sets, MSTW, CTEQ NNPDF (checkother sets) and take central point as uncertainty.
Not very satisfactory, but not clear what would be an improvement, especially as ageneral rule.
Usually not a big disagreement, and factor of about 2 expansion of MSTW uncertainty.
YETI – January 2011 28
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
+H
-HW
+HW
HZ
t (s-channel)
-W
+W
’ (300)+W
’ (600)+W
Z
Z’ (300)
Z’ (600)
tt
H (120)→gg
H (160)→gg
H (250)→gg
t (t-channel)
VBF H (120)
VBF H (160)
VBF H (250)
LHC 7 TeV
CT10CTEQ6.6CT10W
Very Recent Updates
MSTW find new combined HERA datalead to increase in W, Z by couple of %.Less than 1% on Higgs (Tevatron andLHC).
CT10 (right) find change in W, Z verysmall (probably countered by gluonparameterisation change).
Slight increase in Higgs, tt̄ (againprobably gluon shape).
NNPDF find prelim GM-VFNS fits bringthem closer to MSTW,CTEQ for W, Z.
YETI – January 2011 29
Other sources of Uncertainty.
Also other sources which (mainly) lead to inaccuracies common to all fixed-orderextractions.
● QED and Weak (comparable to NNLO ?) (α3s ∼ α). Sometime enhancements.
● Standard higher orders (NNLO – some sets available here.)
● Resummations, e.g. small x (αns lnn−1(1/x)), or large x (αns ln
2n−1(1 − x)) orequivalently summations in high-energy limit and threshold limit.
● low Q2 (higher twist), saturation.
YETI – January 2011 30
NNLO splitting functions now known. (Moch, Vermaseren and Vogt). Essentially fullNNLO determination of partons now being performed (MSTW, ABKM,GJR,HERA),though heavy flavour not fully worked out in the fixed-flavour number scheme (FFNS)PDFs and jet cross-sections approximate. Improve consistency of fit very slightly, andreduces αS.
Surely this is best, i.e. most accurate.
Yes, but ...... only know some hard cross-sections at NNLO.
Processes with two strongly interacting particles largely completed
DIS coefficient functions and sum rules
pp(p̄) → γ?,W, Z (including rapidity dist.), H, A0,WH, ZH.
But for many other final states NNLO not known. NLO still more appropriate.
YETI – January 2011 31
0
0.5
1
1.5
10 -4 10 -3 10 -2 10 -1x
MSTW08 NNLO
MSTW08 NLO
xu(x,Q2=100GeV2)
-10
-5
0
5
10
15
10 -4 10 -3 10 -2 10 -1x
MSTW08 NNLO
MSTW08 NLO
percentage difference at Q2=100GeV2
Stability order-by-order.
Systematic difference betweenPDF defined at NLO and atNNLO.
YETI – January 2011 32
Consideration of NNLO
Very good evidence that one should use NNLO if possible rather than NLO – manyphysical cross-sections, particularly gg → H, not very convergent.
Fewer PDF sets available, can study differences between them better at NLO, but forcentral prediction need NNLO.
Related to issue of use and uncertainty of αS(M2Z). Noted systematic change in value
form fit as one goes from NLO to NNLO. Also highlighted in stability of predictions.
Consider percentage change from NLO to NNLO in MSTW08 predictions for best fitαS compared to fixed αS(M
2Z) = 0.119.
σW (Z) 7TeV σW (Z) 14TeV σH 7TeV σH 7TeVMSTW08 best fit αS 3.0 2.6 25 24MSTW08 αS = 0.119 5.3 5.0 32 30
αS(M2Z) is not a physical quantity. In (nearly) all PDF related quantities (and many
others) shows tendency to decrease from order to order. Noticeable if one has fit atNNLO. Any settling on, or near common αS(M
2Z) has to take this into account.
YETI – January 2011 33
Benchmark results at NNLO
Plots from not yet published results by G. Watt
)2Z
(MSα0.11 0.115 0.12 0.125 0.13
) (
nb
)ν+ l
→ + B
(W⋅ +
Wσ
5.4
5.6
5.8
6
6.2
6.4
6.6
68% C.L. PDF
MSTW08
HERAPDF1.0
ABKM09
GJR08/JR09
= 7 TeV)s at the LHC (ν+ l→ +NNLO W
Open symbols: NLOClosed symbols: NNLO
SαOuter: PDF+Inner: PDF onlyVertical error bars
)2Z
(MSα0.11 0.115 0.12 0.125 0.13
) (
nb
)ν+ l
→ + B
(W⋅ +
Wσ
5.4
5.6
5.8
6
6.2
6.4
6.6
)2Z
(MSα0.11 0.115 0.12 0.125 0.13
-Wσ
/ +Wσ ≡ ±
R
1.4
1.42
1.44
1.46
1.48
1.5
68% C.L. PDF
MSTW08
HERAPDF1.0
ABKM09
GJR08/JR09
= 7 TeV)s ratio at the LHC (-
/W+NNLO W
Open symbols: NLOClosed symbols: NNLO
SαOuter: PDF+Inner: PDF onlyVertical error bars
)2Z
(MSα0.11 0.115 0.12 0.125 0.13
-Wσ
/ +Wσ ≡ ±
R
1.4
1.42
1.44
1.46
1.48
1.5
Differences between PDF sets very similar as at NLO, whereas differences fromtheoretical choices should diminish differences.
More stability if NNLO αS lower than at NLO.
NNLO corrections have minor effect on asymmetries.
YETI – January 2011 34
Plots from not yet published results by G. Watt
)2Z
(MSα0.11 0.115 0.12 0.125 0.13
(p
b)
Hσ
4
4.5
5
5.5
6
6.5
7
68% C.L. PDF
MSTW08
HERAPDF1.0
ABKM09
GJR08/JR09
= 180 GeVH
= 7 TeV) for MsH at the LHC (→NNLO gg
Open symbols: NLOClosed symbols: NNLO
SαOuter: PDF+Inner: PDF onlyVertical error bars
)2Z
(MSα0.11 0.115 0.12 0.125 0.13
(p
b)
Hσ
4
4.5
5
5.5
6
6.5
7
)2Z
(MSα0.11 0.115 0.12 0.125 0.13
(p
b)
ttσ
110
120
130
140
150
160
170
180
190
200
68% C.L. PDF
MSTW08
HERAPDF1.0
ABKM09
GJR08/JR09
= 7 TeV)s cross sections at the LHC (tNNLO (approx.) t
Open symbols: NLOClosed symbols: NNLO
SαOuter: PDF+Inner: PDF onlyVertical error bars
)2Z
(MSα0.11 0.115 0.12 0.125 0.13
(p
b)
ttσ
110
120
130
140
150
160
170
180
190
200
Differences between groups significant at NNLO, and similar to NNLO – partonluminosity compassion similar at NLO to NNLO.
Approx NNLO using HATHOR - (Aliev et al), includes scale-dependent parts andlarge threshold corrections at NNLO. Hence some theoretical uncertainty, but NNLOcorrections not large at LHC.
Top cross-section measurement potential discriminator of PDF sets, and correlated toHiggs predictions.
YETI – January 2011 35
7 TeV 10 TeV 14 TeV
NNLOABM10JRHERAPDFMSTW08
MH
120 GeV
150 GeV
180 GeV
σ(H
0 ) /
pbSimilar study published byAlekhin et al.
Difficult to compare PDFuncertainties meaningfully inthis plot, but size of scaleuncertainty illustrated for somesets.
Dominates Higgs cross-section,but very highly correlatedbetween PDFs, i.e. overlapin uncertainties when includednot strictly “agreement”.
Consider full PDF uncertaintyand then theory (scale)uncertainty separately. Correlationdepends on process, buthopefully low.
YETI – January 2011 36
Differences in rapidity distributions evident at NNLO.
Based on (Anastasiou et al.), from Lance Dixon.
Differences bigger than uncertainties.
YETI – January 2011 37
Electroweak corrections
Typically a few percent, e.g. Calone Calame et al who look at Drell-Yan processes.
Also consider photon-induced processes. Requires the photon distribution of theproton. Currently only one QED-corrected pdf (MRST2004) set (leads to automaticisospin violation - reduces NuTeV anomaly).
Can also be a couple of percent (here in opposite direction).
YETI – January 2011 38
Large Electroweak corrections
Jet cross-section a major example – calculation by Moretti, Nolten, Ross, goes like(1 − 13CF
αWπ
log2(E2T/M2W )).
Big effect at LHC energies – log2(E2T/M2W ) a very large number. Up to 30%. Bigger
than NLO QCD.
YETI – January 2011 39
( � � � Born)
� Born
PcutT (l� l̄� ) [GeV]700650600550500450400350300250
0
� 0� 05
� 0� 1
� 0� 15
� 0� 2
� 0� 25
� 0� 3
� 0� 35
� 0� 4
Similar results for correctionsto other processes with a hardscale, e.g. Di-boson production(Accomando et al).
Plot shows fractional correctionsas function of reconstructed Ztransverse momentum in WZproduction.
Same sort of corrections in large-pT vector bosons in conjunctionwith jets (Kühn et al, Maina etal)...
ln(s/m2W ) terms can also affectΓW extraction from the transversemass distribution.
YETI – January 2011 40
[GeV]HM100 200 300 400 500 600 700 800 900 1000
qqH
) [p
b]→
(pp
σ
-210
-110
1 NLO QCD+EW - 7 TeV
NLO QCD - 7 TeV
LH
C H
IGG
S X
S W
G 2
010
[GeV]HM100 200 300 400 500 600 700 800 900 1000
qqH
) [p
b]→
(pp
σ
-110
1
NLO QCD+EW - 14 TeV
NLO QCD - 14 TeV
LH
C H
IGG
S X
S W
G 2
010
Effect of electroweak corrections to Higgs production from vector boson fusion.
Plots from Vector Boson Fusion section of Handbook of LHC Higgs Cross Sections.
YETI – January 2011 41
Small-x Theory
At each order in αS each splitting function and coefficient function obtains an extrapower of ln(1/x) (some accidental zeros in Pgg), i.e. Pij(x, αs(Q
2)), CPi (x, αs(Q2)) ∼
αms (Q2) lnm−1(1/x).
Summed using BFKL equation (and a lot of work – Altarelli-Ball-Forte, Ciafaloni-Colferai-Salam-Stasto and White-RT)
10 210
0
0.5
1
= 460, 575, 920 GeVpE
0.00
0059
0.00
0087
0.00
013
0.00
017
0.00
021
0.00
029
0.00
040
0.00
052
0.00
067
0.00
090
0.00
11
0.00
15
0.00
23x
H1 (Prelim.) MSTW NLO MSTW NNLO WT NLO + NLL(1/x)
LH1 Preliminary F
2 / GeV 2Q
)2 (
x, Q
LF
0
0.5
1
10 210
Comparison to H1 prelim data onFL(x,Q
2) at low Q2, only withinWhite-RT approach, suggestsresummations may be important.
Could possibly give a few percenteffect on Higgs cross sections.
YETI – January 2011 42
y0 1 2 3 4 5 6 7 8 9 10
/dM
/dy
(p
b/G
eV)
σ2 d
-2
0
2
4
6
8
10
12
14
163
10×
M = 2 GeVM = 2 GeV
y0 1 2 3 4 5 6 7 8 9 10
/dM
/dy
(p
b/G
eV)
σ2 d
0
0.5
1
1.5
23
10×
M = 4 GeVM = 4 GeV
y0 1 2 3 4 5 6 7 8 9 10
/dM
/dy
(p
b/G
eV)
σ2 d
-50
0
50
100
150
200
M = 10 GeV
(2001 LO, 2004 NLO, 2006 NNLO)Using Vrap with MRST PDFs
M = 10 GeV
y0 1 2 3 4 5 6 7 8 9 10
/dM
/dy
(p
b/G
eV)
σ2 d
-20
0
20
40
60
80
100
ZM = M
LO
NLO
qNLO q
NLO qg
NNLO
qNNLO q
NNLO qg
NNLO gg
NNLO qq
ZM = M
*/Z rapidity distributions at LHCγThe region of large theory correctionsis lower M2 and high y.
However, this assumes perturbativeprediction of Drell-Yan production isreliable.
As seen very large change inprediction from order to order,particularly for low M and high y.
Problem with perturbative stability.Is this due to partons or cross-sections?
YETI – January 2011 43
y0 1 2 3 4 5 6 7 8 9 10
/dM
/dy
(p
b/G
eV)
σ2 d
-2
0
2
4
6
8
10
12
14
163
10×
M = 2 GeVM = 2 GeV
y0 1 2 3 4 5 6 7 8 9 10
/dM
/dy
(p
b/G
eV)
σ2 d
0
0.5
1
1.5
23
10×
M = 4 GeVM = 4 GeV
y0 1 2 3 4 5 6 7 8 9 10
/dM
/dy
(p
b/G
eV)
σ2 d
-50
0
50
100
150
200
M = 10 GeV
MRST 2006 NNLO PDFsUsing Vrap with
M = 10 GeV
y0 1 2 3 4 5 6 7 8 9 10
/dM
/dy
(p
b/G
eV)
σ2 d
-20
0
20
40
60
80
100
ZM = M
LO
NLO
qNLO q
NLO qg
NNLO
qNNLO q
NNLO qg
NNLO gg
NNLO qq
ZM = M
*/Z rapidity distributions at LHCγKeeping partons fixed while changingcross-sections (using MRST2006NNLO partons) also shows part ofinstability due to partons. Unusualbehaviour in very small x partons atNNLO. Due to similar high and lowz terms in splitting functions.
Overall most obvious effect – largechange in quark-gluon (and quark-quark) contributions at NNLO dueto 1/z and ln(1 − z) divergences incross-sections appearing at this order.
Cross-section may be sensitive toresummations (high and low z) atlowest M and highest y. In regionwhere measurements can be made?
YETI – January 2011 44
Conclusions
We can calculate to NLO or NNLO in QCD and sometimes include electroweakcorrections. Also need PDFs.
One can determine the parton distributions and predict cross-sections at the LHC, andthe fit quality using NLO or NNLO QCD is fairly good.
Various ways of looking at experimental uncertainties on PDFs. Uncertainties∼ 1 − 5% for most LHC quantities. Major uncertainty in vector boson production.Ratios, e.g. W+/W− tight, and hopefully early constraint on partons.
Effects from input assumptions e.g. selection of data fitted, cuts and inputparameterisation can shift central values of predictions significantly. Also affect sizeof uncertainties. Want balance between freedom and sensible constraints. Completeheavy flavour treatments essential in extraction and use of PDFs. αS and PDFsheavily correlated.
Errors from higher orders estimated using scale variations, which should often bereasonable. Can dominate, e.g. Higgs. Resummation effects also potentially large.At LHC measurement at high rapidities, e.g. W, Z would be useful in testingunderstanding of QCD, and particularly quantities sensitive to low x at low scales,e.g. low mass Drell-Yan.
YETI – January 2011 45
Comparison to Standard Model predictions at the LHC far from a straightforwardprocedure. Lots of theoretical issues to consider for real precision. Relatively fewcases where Standard Model discrepancies will not require some significant input fromQCD, PDF and electroweak physics to determine real significance.
YETI – January 2011 46
However, does include pre-processing exponents as x → 1 and x → 0 to aidconvergence of fit,
f(x,Q20) = A(1 − x)mx−nNN(x)
where n,m are in fairly narrow ranges, so overall behaviour guided at these extremeswhere data constraints vanish.
GA generations0 5000 10000 15000 20000 25000 30000 35000 40000 45000 500001
1.5
2
2.5
3
3.5
NMC-pd
trE valE
targetE
NMC-pdSplit data sets randomly intoequal size training and validationsets.
Fit until quality of fit to validationset starts to go up, even thoughtraining set still (hopefully slowly)improving.
Criterion for stopping the fitdepends on different data sets.
Uncertainty has depended on stopping criteria.
YETI – January 2011 47
x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
) 02xV
(x,
Q
0
0.2
0.4
0.6
0.8
1
1.2 NNPDF2.0CTEQ6.6
MSTW 2008
Uncertainties on, e.g. valence quarks not notably different to other groups at all.
YETI – January 2011 48
0.500 TeV 0.546 TeV 0.630 TeV 1.8 TeV 1.96 TeV
NNLOABM10JRHERAPDFMSTW08
UA1UA2PHENIX W +
PHENIX W -
CDFD0
W + + W -
W + + W -
Z 0
Z 0
σ(V
) / n
b
(a)
Study published by Alekhin etal.
Note consistent normalisationdifference between CDF andD0.
YETI – January 2011 49
Small-x Theory
Reason for this instability – at each order in αS each splitting function and coefficientfunction obtains an extra power of ln(1/x) (some accidental zeros in Pgg), i.e.Pij(x, αs(Q
2)), CPi (x, αs(Q2)) ∼ αms (Q2) lnm−1(1/x).
BFKL equation for high-energy limit
f(k2, x) = fI(Q20)+
∫ 1
xdx′
x′ᾱS∫∞
0dq2
q2K(q2, k2)f(q2, x),
where f(k2, x) is the unintegrated gluon distribution
g(x, Q2) =∫ Q2
0(dk2/k2)f(x, k2), and K(q2, k2) is a
calculated kernel known to NLO.
Physical structure functions obtained from
σ(Q2, x) =∫
(dk2/k2)h(k2/Q2)f(k2, x)
where h(k2/Q2) is a calculable impact factor.
The global fits usually assume that this is unimportantin practice, and proceed regardless.
Fits work well at small x, but could improve.
YETI – January 2011 50
-3
-2
-1
0
1
2
10 -5 10 -4 10 -3 10 -2 10 -1 1x
Q2=1GeV2xg
(x)
0
20
40
60
80
10 -5 10 -4 10 -3 10 -2 10 -1 1x
Q2=100GeV2
NLL+NLL(2)+NLO+
Good recent progress in incorporatingln(1/x) resummation Altarelli-Ball-Forte, Ciafaloni-Colferai-Salam-Stastoand White-RT.
Include running coupling effects andvariety (depending on group) of othercorrections
By 2008 very similar results comingfrom the competing procedures,despite some differences in technique.
Full set of coefficient functions stillto come in some cases, but splittingfunctions comparable.
Note, in all cases NLO correctionslead to dip in functions below fixedorder values until slower growth(running coupling effect) at verysmall x.
YETI – January 2011 51
A fit to data with NLO plus NLO resummation, with heavy quarks included (White,RT)performed.
0
0.5
1
1.5
2
2.5
3
3.5
4
1 10 10 2 10 3
x=5×10-4
x=6.32×10-4
x=8×10-4
x=1.3×10-3
x=1.61×10-3
x=2×10-3
x=3.2×10-3
x=5×10-3
x=8×10-3
H1ZEUSNMC
NLL+NLL(2)+NLO+
Q2(GeV2)
F 2p (x
,Q2 )
+ 0.
25(9
-i)
-3
-2
-1
0
1
2
10 -5 10 -4 10 -3 10 -2 10 -1 1x
Q2=1GeV2xg
(x)
0
20
40
60
80
10 -5 10 -4 10 -3 10 -2 10 -1 1x
Q2=100GeV2
NLL+NLL(2)+NLO+
→ moderate improvement in fit to HERA data within global fit, and change inextracted gluon (more like quarks at low Q2).
Together with indications from Drell Yan resummation calculations (Marzani, Ball)few percent effect quite possible.
YETI – January 2011 52