1. Parton Uncertainties and the Stability of NLO Global Analysis
Daniel Stump
Department of Physics and Astronomy
Michigan State University
J. Huston, J. Pumplin, D. Stump and W.K. Tung,
Stability of NLO Global Analysis and Implications for Hadron Collider Physics,
hep-ph/
The title of my talk is {READ}.
I will mainly talk about this recent paper by members of CTEQ at Michigan State University.
4/29/2005
DIS 2005

2. 1. Global Analysis and Uncertainties 2. The CTEQ stability study
Outline:
1. Global Analysis and Uncertainties
2. The CTEQ stability study
3. The Lagrange Multiplier method and the gluon distribution function
The outline of the talk is first some general comments about Global Analysis and Uncertainties; second the CTEQ stability study; and third the use of the Lagrange Multiplier method in this study.
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DIS 2005

3. 1/ Global Analysis of QCD …
…uses data from many experiments to construct universal parton distribution functions (PDF’s).
Tevatron
(CDF and D0)
HERA
(H1 and ZEUS)
BCDMS m p, m d
CCFR n Fe
E605 pp, pd
E866 pp, pd
{READ 1}
For example, data from DIS --- the extensive data from HERA and other experiments --- is crucial. Data from the Drell-Yan process is complementary. And, inclusive jet production at the Tevatron constrains the gluon distribution function.
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4. … based on features of QCD • Factorization theorem
• Infrared safety of inclusive cross sections
• Asymptotic freedom (e.g., to justify the use of NLO perturbation theory)
So, by combining experimental data from hadron collisions, and theoretical calculations of parton cross sections we construct the PDF’s….
{READ}
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5. of Parton Distribution Functions
Uncertainties
of Parton Distribution Functions
Now, let me make some general comments about uncertainties of parton distribution functions.
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6. U-quark Q2 = 10 GeV2 Gluon We cannot determine exactly the PDF’s.
Rather, we fit simultaneously the chosen global data sets, within a range of uncertainty.
These graphs show alternate fits — the 40 eigenvector basis sets — for CTEQ6.1. They illustrate the uncertainties of the PDF’s. The black curve is the standard fit, at the minimum of c2. But we cannot rule out any of the alternate fits based on the current data; they have only slightly higher c2.
For the U-quark the uncertainty is quite small. {Point}
The Gluon has a larger uncertainty. {Point}
Gluon
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7. Blue:
MRST2002
Red:
MRST2003c
The gluon uncertainty is illustrated again in this figure.
The graph shows g(x) divided by CTEQ6.1 as a function of x on a log scale.
The green shaded region is the uncertainty range, of roughly +/- 10 % for small x; larger for large x.
The curves are 2 MRST gluon distributions {READ legend}. These are comparable to CTEQ6 but which lie below CTEQ6 at both small x and large x.
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8. The question of compatibility
Are data sets from different experiments compatible?
The question of stability
Are the final results of the global analysis stable and robust?
In Global Analysis, we inevitably face two questions.
{READ}
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DIS 2005

9. Compatibility
Collaboration A
Collaboration B
Two experimental collaborations measure the same quantity q :
Compatibility!
The question is, are different data sets compatible?
Let me discuss this question in general terms with a little example.
Suppose 2 experimental collaborations (A and B) measure the same quantity theta, with these results. Each experiment has statistical errors and systematic errors. And because of the errors, there is a systematic difference between the results.
{READ 1}
Now, suppose theorists need the value of theta for predictions for future experiments. What value should they use? Most likely they would try to combine the results of A and B. In an ideal world of perfect Gaussian errors, there would be a well-defined procedure for combining the results, including the systematic errors. But in the real world, where systematic errors may be non-Gaussian, some subjective judgment will be required. But in any case,
{READ 2}
The two data sets are consistent within the systematic errors, but there is a systematic difference.
The combined value is a compromise, with uncertainty from the systematic errors.
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10. PDF’s from global analysis are a compromise, with uncertainties from systematic errors.
Are data sets compatible?
The only way to compare different processes, e.g., DIS and pp jet production, is through the global analysis. 
Because of systematic errors, we do find minor incompatibilities: The best fit to one data set is not the best fit to another data set. Nevertheless, all data sets can be fit simultaneously within the systematic errors. 
That simple example is exactly what happens in Global Analysis of QCD.
{READ 1}
However, the situation is more complex in Global Analysis, because the experimental collaborations are measuring different quantities.
{READ rest}
So, we must expect, and accept, that the final PDF’s are a compromise --- and not the best possible fit to any specific experiment.
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11. 2/ The question of stability
Minor changes of inputs (data and theory choices) should not produce large changes of the PDF’s.
But the stability of NLO global analysis has been challenged by an interesting result of the MRST group.
Reference:
Martin, Roberts, Stirling and Thorne, Eur Phys J C35, 325 (2004).
Remove low-Q and low-x data; i.e., require
Q > Qcut and x > xcut.
Are the resulting PDF’s stable with respect to changes of Qcut and xcut?
Next, the question of stability.
In order for the results of a global analysis to be trustworthy, they must be robust.
That is, {READ 1}.
{READ 2} - reference to an important paper on theoretical errors on PDF’s.
{APPEAR and READ 3}
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DIS 2005

12. A surprising “instability” of sW(LHC) in the NLO approx’n.
Reference: Martin, Roberts, Stirling, Thorne, Eur. Phys. J. C35, 325 (2004)
The MRST group found…… {READ 1}.
Focus on the circles, which are the NLO results for the cross section sW as a function of the x-cut.
The default is x-cut = 0 for which sW is about 20 nb. But as x-cut is increased, the cross section goes down. For x-cut of 0.005, sW is about 16 nb, smaller by 20%.
Evidently as the low-x data is removed, other data (especially the Tevatron jet data) pull the PDF’s away from the default fform, which yields a different prediction for sW. MRST suggested that this represents a large theoretical uncertainty for the NLO global analysis. Indeed, the prediction appears to be unstable.
{APPEAR 2}
“default”
“conservative”
Is the instability a breakdown of NLO QCD, or a consequence of PDF uncertainties, or an artifact of the parameterization?
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13. CTEQ stability study
First step – apply the MRST procedure to the CTEQ parameterization of PDF’s
Cuts
Qmin
xmin
Npts
c21926
c21770
c21588
sW.B
standard
2 GeV
1926
2023
1850
1583
20.02
intermed
2.5 “
0.001
1770 --
1849
1579
20.10
strong
3.16 “
0.005
1588
1573
20.34
[nb]
TABLE 1: The best fits for three choices of exclusionary cuts (standard, intermediate and strong) with a positive-definite parameterization of the gluon PDF.
Well, the CTEQ group at Michigan State has undertaken a similar test of the stability of the NLO analysis.
{READ 1}
The table shows {Read Table caption}.
The standard cuts are approximately the MRST “default cuts”, and the strong cuts are approximately the MRST “conservative cuts”.
Npts is the number of data points for standard cuts, reduced to 1588 for strong cuts.
Of course the total c2 decreases with the number of points. But the interesting question is how c2 for the subset of data that pass the strong cuts changes as the x-cut increases. The c2 for that subset of data is shown in this column; it changes very little as x-cut is increased, indicating that the NLO global analysis is stable.
Also the LHC cross section sW is stable; it changes by less than the PDF uncertainty.
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14. allowing g(x) < 0 (for small x and Q) …
CTEQ stability study
allowing g(x) < 0 (for small x and Q) …
Cuts
Qmin
xmin
Npts
c21926
c21770
c21588
sW.B
standard
2 GeV
1926
2011
1845
1579
19.94
intermed
2.5 “
0.001
1770 --
1838
1574
19.80
strong
3.16 “
0.005
1588
1570
19.15
[nb]
TABLE 2: The best fits for three choices of exclusionary cuts (standard, intermediate and strong) with an extended parameterization that allows g(x) < 0.
We also considered extending the parameterization so that the gluon distribution function is allowed to be negative for small x and Q, in case that would significantly improve the fit to data. A negative gluon has been a feature of the MRST gluon distribution.
But again we find that the NLO global analysis is stable with respect to the x-cut. Again, the cross section sW changes very little.
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15. CTEQ stability study Results, graphically:
Here are the results graphically.
The black dots are the MRST results for NLO.
The red crosses are CTEQ with a postive-definite gluon, and the blue crosses are CTEQ allowing the gluon PDF to go negative at small x.
For either CTEQ case, making the strong cuts does not change significantly the value of sW.
The predicted total cross section of W++W- production at the LHC, for NLO calculations.
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16. Stability of the extracted value of aS
The global fit c2 as a function of aS(MZ)
Here is another example of stability of the NLO global analysis --- in the extracted value of aS.
We choose a value of aS (at the Z mass) as an input to the analysis.
If we vary the choice of aS, the fit will change.
The figure shows how the global c2 varies with aS.
{READ legend}
From the c2 parabola, we extract the true value of aS at the minimum, with an error band from the increase of c2.
Black: positive-definite g(x);
Blue: negative gluon is allowed.
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17. Cuts positive gluon g(x) < 0 allowed Standard 0.1169 ± 0.0045
Extracted values of
Cuts
positive gluon
g(x) < 0 allowed
Standard ± ±
Strong ± ±
i.e., the extracted value of aS is clearly stable with respect to x- and Q- cuts on data.
Positive definite gluon – stable wrt cuts.
Negative gluon allowed – stable wrt cuts.
There is a small difference between the 2 parameterizations, but less than the uncertainties.
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18. 3/ The Lagrange Multiplier method
To gain more insight into the results…
… probe the uncertainty of a prediction from the global analysis
using the Lagrange Multiplier method.
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19. The differential cross section, ds/dy
The differential cross section, ds/dy. [MRST, Eur Phys J C35, 325 (2004)]
MRST paper: Removing the constraints of data with x < radically changes the NLO PDF’s and hence the cross section for W production.
Here again is the cross section for W production at the LHC.
The thumbnail graph shows the total cross section. The large graph is the MRST calculation of the differential cross section ds/dy for the “default” and “conservative” cuts.
The “default cuts” correspond to this point, and the “conservative cuts” to this point.
For the “conservative” cuts, ds/dy falls rapidly with yW. This explains why the total cross section is much smaller.
With the strong (or, conservative) cuts, the negative gluon pulls the cross section down at high rapidity, and the prediction looks very different.
{READ (MRST paper)}
But this analysis does not go far enough, because it does not show how the strong (or, conservative) cuts affect the uncertainty of the prediction.
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20. Lagrange Multiplier method  calculate c2 versus sW.
Black curve:
standard cuts (xmin=0)
Blue curve:
strong cuts (xmin=0.005)
The effects of the strong cuts:
 the central prediction barely moves;
 the uncertainty increases significantly.
{READ 1}
The striking feature is not the small shift in the minimum, but the large broadening of the c2 parabola.
So, we now see the effects of the strong cuts:
{APPEAR and READ 2}
If there is a +/- 5% uncertainty with standard cuts, that broadens to a +/- 10% uncertainty with strong cuts.
[positive gluon]
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21. Tevatron: W production can occur by a LO process with valence quarks.
W production at the LHC is sensitive to the gluon distribution function.
Tevatron: W production can occur by a LO process with valence quarks.
LHC: The LO contribution must involve a sea quark; and there is an NLO contribution from a gluon.
It is easy to see that ……
{READ rest}
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22. Green: Extremes of 40 EV sets
The Gluon
distribution
Black: CTEQ6.1
Green: Extremes of 40 EV sets
Blue: MRST2002 (default)
Red: MRST2003c (conservative)
This figure indicates the contrast between the MRST and CTEQ parameterizations at small x.
{READ legend}
The default MRST gluon distribution goes negative at very small x. /6E-4/
The conservative PDF goes negative at a larger value of x. /4E-3/
Removing the constraints of the small-x data gives a better fit to the Tevatron jet data, but at the expense of a more negative gluon in the MRST parameterization. That pulls down the LHC W production.
The CTEQ parameterization, in contrast, fits both Tevatron data and DIS data well for the standard cuts. In other words, the strong cuts are not necessary to allow the parameterization to fit the Tevatron data.
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23. Red: MRST2003c (conservative)
Sea quarks:
u-bar distribution
Black: CTEQ6.1
Blue: MRST2002 (default)
Red: MRST2003c (conservative)
The sea quark distributions are also affected in the MRST fits --- being pulled down if the small-x data is cut out.
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24. Green: extremes of the 40 eigenvector basis sets
The Gluon PDF at large x
Black: CTEQ6.1
Green: extremes of the 40 eigenvector basis sets
Blue: MRST2002
Red: MRST2003c (conservative)
Violet: MRST2004 (physical)
There is still a lot of uncertainty in the gluon PDF. This graph shows g(x) for large Q, focusing on large x.
{READ CTEQ legend}
More data will be needed to pin down the gluon to make precise LHC predictions.
{READ MRST legend}
Here we see how the MRST gluon at large x lies below the CTEQ6 gluon. The violet curve is the most recent MRST parameterization, dubbed the “Physical Gluon”, which fits the Tevatron jet data better.
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25. Conclusions For the CTEQ parameterization …
 The NLO global analysis is stable with respect to cuts on x and Q. (A strong cut on x is not needed to fit DIS and Tevatron data simultaneously; and it would increase the uncertainty.)
 A positive-definite gluon parameterization is satisfactory.
 Additional data will be needed to constrain the gluon PDF for accurate LHC predictions.
4/29/2005
DIS 2005