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# PDF Contribution to UEH Systematic: CTEQ Method

Updated on Thu, 2016-07-21 21:55. Originally created by pagebs on 2016-07-15 10:00.

Here I recalculate the PDF uncertainty contribution to the UEH correction for the cross section using the CTEQ 'Master Formula' instead of just taking an envelope ...

The error sets which come with a CTEQ PDF central set each represent the variation in one of the PDF fitting parameters and come in pairs, one for a positive and one for a negative change. I use the CTEQ6m PDF set to calculate the PDF contribution to the UEH systematic. This set has 40 associated error sets so there were 20 varried parameters. CTEQ has a method for estimating the error on a given observable due to PDF errors:

Delta_X = 0.5*[Sum_i(X[S_i^+] - X[S_i^-])^2]^(1/2)

where X is the observable (the ratio of the particle/no mass parton level yield), X[S_i^+] is the value of the observable using the ith positive eigenvector error set, X[S_i^-] is the value of the observable using the ith negative eigenvector error set, and the sum over i runs over the 20 error sets.

There is also an asymmetric formulation:

Delta_X+ = SQRT{Sum_i(MAX[X[S_i^+] - X[S_0], X[S_i^-] - X[S_0], 0)^2}

Delta_X- = SQRT{Sum_i(MAX[X[S_0] - X[S_i^+], X[S_0] - X[S_i^-], 0)^2}

where the new term, X[S_0], is the value of the observable using the best fit PDF.

Figure 1: Ratio of the error on the particle/no mass parton ratio using my original method in which I took the distance between the central value curve and the edge of the envelope formed by all 40 error sets as the systematic (black), the symmetric CTEQ method (red) and the asymmetric CTEQ method (blue). The x-axis is bin number and shows only the 9 mass bins used in the final cross section.

As can be seen there are several bins with large excursions. To get a better idea of what is causing these, I printed out the contributions from all 20 error sets to the total error for each mass bin. They are in this text file.

I also plotted the particle/no mass parton ratio for each of the 20 error set pairs along with the CTEQ6m central value and CTEQ5L (which was used in the official simulation) central value. The plots for each of the 20 error set pairs can be seen in the attached files section at the bottom of the page.

As an example, lets look at the 7th bin in figure 1. If I look at the section under 'Bin 6' in the text file, I see that for both the symmetric case and the positive asymmetric case, there is one error set which is dominating the contribution to the systematic which is Bin 18. This corresponds to error set 19 (naming schemes are off by 1) and if we look at that figure, we see a large excursion in the positive value of the error set in the 7th used bin (remember we don't show the first or last mass bin). This is shown in figure 2.

Figure 2: Ratio of the particle/no mass parton yields for the CTEQ5L (blue), CTEQ6m central value (red) and positive and negative error sets associated with the 19th eigenvector (black). A large excursion can be seen in the 7th used mass bin.

Looking through the text file and the rest of the error set plots, there are several outliers which I have removed from the calculation of the uncertainty on the UEH:

Error sets 2 and 18 were removed in the calculation of the error in the third used mass bin

Error set 9 was removed in the calculation of the error in the fourth used mass bin

Error set 19 was removed in the calculation of the error in the seventh used mass bin

It also appears that error set 5 is the dominant contribution to the uncertainty in all mass bins, but that is kept for now.

After the above outliers were removed, I calculated the error/central value curves again:

Figure 3: Same as figure 1, but after outlying error sets were removed

As can be seen from figure 3, my naieve method for calculating the PDF uncertainty contribution to the UEH systematic is pretty close to the 'correct' CTEQ method. I will use the symmetric errors (red curve) for my final plots. These errors are combined with the factorization/renormalization errors to form the full UEH systematic, and the full UEH systematic is combined with the systematic on the actual theory values to form the final overall theory systematic, so changes to the systematic values / plot will be negligible.

I believe that the deviation to the low side in the asymmetric method is due to the fact that the actual central value fluctuated positive.

Looking at the comparisons between the error sets and central value for the particle/no mass parton level yields, it seems like error set 5 has the largest deviation, although some other sets show non-negligible deviations. To get a better idea of what is driving the disagreement, look at the light quark and gluon PDFs for the central and two variations sets associated with error set 5. First I show the central values for all curves at Q=2 GeV to verify I am reading the files correctly.

Figure 4: Dbar, Ubar, Gluon, U, and D x*PDF vs x at an input scale of 2 GeV.

Figure 5: Comparison of CTEQ6m central value and error set 5 (PDF sets 9 and 10) gluon PDFs. The top panel shows the PDFs at Q=1.3GeV, the middle panel shows Q=2GeV, and the bottom panel shows Q=5GeV.

Figure 6: Same as figure 5, but showing the dbar PDFs for comparison.

It seems that differences in the gluon distribution are mainly responsible for the differences seen in error set 5, although they become smaller at higher Q. Comparisons from the remaining PDFs can be found in the files below starting with cteqErr_*.

Figure 7: Similar to figure 2. The nominal value of the pythia/no mass parton ratio is shown along with the central value using CTEQ6m and the positive and negative variations (pdf sets 9 and 10) associated with the fifth error set.

The contribution of the PDF error to the underlying event and hadronization correction has been found using the symmetric CTEQ method. The size of the error can be seen in the red curve of figure 3. Looking through all 20 error sets, much of the contribution comes from set 5. To see what causes the deviation, I plotted the parton densities for the CTEQ6m central value and error sets 9 and 10 (the sets associated with error set 5). The gluon density in error set 10 is higher at low x and lower at high x and vis versa for error set 9. We see in figure 7 that the particle over parton ratio is higher for set 10 meaning that the low x gluon distribution dominates. The low x region contributes to the underlying event, so the pdf error contribution presented here probes this. However, the perugia0 tune assumes a specific PDF (which is not CTEQ6m), so this error is likely conflating the affects from various pythia parameters as well as the actual contributin from the PDF and is therefore maybe overestimating the PDF error. This would take too long to explain in the paper so the text will remain the same.

The error sets which come with a CTEQ PDF central set each represent the variation in one of the PDF fitting parameters and come in pairs, one for a positive and one for a negative change. I use the CTEQ6m PDF set to calculate the PDF contribution to the UEH systematic. This set has 40 associated error sets so there were 20 varried parameters. CTEQ has a method for estimating the error on a given observable due to PDF errors:

Delta_X = 0.5*[Sum_i(X[S_i^+] - X[S_i^-])^2]^(1/2)

where X is the observable (the ratio of the particle/no mass parton level yield), X[S_i^+] is the value of the observable using the ith positive eigenvector error set, X[S_i^-] is the value of the observable using the ith negative eigenvector error set, and the sum over i runs over the 20 error sets.

There is also an asymmetric formulation:

Delta_X+ = SQRT{Sum_i(MAX[X[S_i^+] - X[S_0], X[S_i^-] - X[S_0], 0)^2}

Delta_X- = SQRT{Sum_i(MAX[X[S_0] - X[S_i^+], X[S_0] - X[S_i^-], 0)^2}

where the new term, X[S_0], is the value of the observable using the best fit PDF.

Figure 1: Ratio of the error on the particle/no mass parton ratio using my original method in which I took the distance between the central value curve and the edge of the envelope formed by all 40 error sets as the systematic (black), the symmetric CTEQ method (red) and the asymmetric CTEQ method (blue). The x-axis is bin number and shows only the 9 mass bins used in the final cross section.

As can be seen there are several bins with large excursions. To get a better idea of what is causing these, I printed out the contributions from all 20 error sets to the total error for each mass bin. They are in this text file.

I also plotted the particle/no mass parton ratio for each of the 20 error set pairs along with the CTEQ6m central value and CTEQ5L (which was used in the official simulation) central value. The plots for each of the 20 error set pairs can be seen in the attached files section at the bottom of the page.

As an example, lets look at the 7th bin in figure 1. If I look at the section under 'Bin 6' in the text file, I see that for both the symmetric case and the positive asymmetric case, there is one error set which is dominating the contribution to the systematic which is Bin 18. This corresponds to error set 19 (naming schemes are off by 1) and if we look at that figure, we see a large excursion in the positive value of the error set in the 7th used bin (remember we don't show the first or last mass bin). This is shown in figure 2.

Figure 2: Ratio of the particle/no mass parton yields for the CTEQ5L (blue), CTEQ6m central value (red) and positive and negative error sets associated with the 19th eigenvector (black). A large excursion can be seen in the 7th used mass bin.

Looking through the text file and the rest of the error set plots, there are several outliers which I have removed from the calculation of the uncertainty on the UEH:

Error sets 2 and 18 were removed in the calculation of the error in the third used mass bin

Error set 9 was removed in the calculation of the error in the fourth used mass bin

Error set 19 was removed in the calculation of the error in the seventh used mass bin

It also appears that error set 5 is the dominant contribution to the uncertainty in all mass bins, but that is kept for now.

After the above outliers were removed, I calculated the error/central value curves again:

Figure 3: Same as figure 1, but after outlying error sets were removed

As can be seen from figure 3, my naieve method for calculating the PDF uncertainty contribution to the UEH systematic is pretty close to the 'correct' CTEQ method. I will use the symmetric errors (red curve) for my final plots. These errors are combined with the factorization/renormalization errors to form the full UEH systematic, and the full UEH systematic is combined with the systematic on the actual theory values to form the final overall theory systematic, so changes to the systematic values / plot will be negligible.

I believe that the deviation to the low side in the asymmetric method is due to the fact that the actual central value fluctuated positive.

Looking at the comparisons between the error sets and central value for the particle/no mass parton level yields, it seems like error set 5 has the largest deviation, although some other sets show non-negligible deviations. To get a better idea of what is driving the disagreement, look at the light quark and gluon PDFs for the central and two variations sets associated with error set 5. First I show the central values for all curves at Q=2 GeV to verify I am reading the files correctly.

Figure 4: Dbar, Ubar, Gluon, U, and D x*PDF vs x at an input scale of 2 GeV.

Figure 5: Comparison of CTEQ6m central value and error set 5 (PDF sets 9 and 10) gluon PDFs. The top panel shows the PDFs at Q=1.3GeV, the middle panel shows Q=2GeV, and the bottom panel shows Q=5GeV.

Figure 6: Same as figure 5, but showing the dbar PDFs for comparison.

It seems that differences in the gluon distribution are mainly responsible for the differences seen in error set 5, although they become smaller at higher Q. Comparisons from the remaining PDFs can be found in the files below starting with cteqErr_*.

Figure 7: Similar to figure 2. The nominal value of the pythia/no mass parton ratio is shown along with the central value using CTEQ6m and the positive and negative variations (pdf sets 9 and 10) associated with the fifth error set.

The contribution of the PDF error to the underlying event and hadronization correction has been found using the symmetric CTEQ method. The size of the error can be seen in the red curve of figure 3. Looking through all 20 error sets, much of the contribution comes from set 5. To see what causes the deviation, I plotted the parton densities for the CTEQ6m central value and error sets 9 and 10 (the sets associated with error set 5). The gluon density in error set 10 is higher at low x and lower at high x and vis versa for error set 9. We see in figure 7 that the particle over parton ratio is higher for set 10 meaning that the low x gluon distribution dominates. The low x region contributes to the underlying event, so the pdf error contribution presented here probes this. However, the perugia0 tune assumes a specific PDF (which is not CTEQ6m), so this error is likely conflating the affects from various pythia parameters as well as the actual contributin from the PDF and is therefore maybe overestimating the PDF error. This would take too long to explain in the paper so the text will remain the same.

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