We calculate the first and second moments of the photon spectrum. We use these moments to calculate the mean and variance of the spectrum, quantities that are used as input to different models of the photon spectrum. We calculate the quantities:
Mn= P18
i=1BFiEγin P18
i=1BFi
, (11.3)
Table11.6:Correlationcoefficientforthesystematicerrorsbetweenmassbins. mXs0.6-0.70.7-0.80.8-0.90.9-1.01.0-1.11.1-1.21.2-1.31.3-1.41.4-1.51.5-1.61.6-1.71.7-1.81.8-1.91.9-2.02.0-2.22.2-2.42.4-2.62.6-2.8 0.6-0.71.0000.8510.4850.5320.3630.5610.5110.4900.4210.4320.3730.3560.6610.3550.1500.0890.060-0.166 0.7-0.80.8511.0000.8630.8670.4190.6820.6810.6650.5490.5400.4640.4260.5790.3670.2240.1300.089-0.162 0.8-0.90.4850.8631.0000.9350.3780.6680.7110.7010.5610.5130.4470.4010.3080.2870.2480.1430.099-0.148 0.9-1.00.5320.8670.9351.0000.3820.7090.7540.7430.5960.5490.4770.4290.3500.3120.2630.1520.105-0.155 1.0-1.10.3630.4190.3780.3821.0000.3600.3610.3520.2870.2590.2300.2140.2080.1680.1180.0690.047-0.104 1.1-1.20.5610.6820.6680.7090.3601.0000.9940.9900.9640.5490.5380.5070.3500.3930.3560.2340.181-0.253 1.2-1.30.5110.6810.7110.7540.3610.9941.0000.9990.9540.5710.5660.5320.3350.4070.3890.2580.203-0.257 1.3-1.40.4900.6650.7010.7430.3520.9900.9991.0000.9580.5780.5830.5490.3460.4260.4180.2810.225-0.260 1.4-1.50.4210.5490.5610.5960.2870.9640.9540.9581.0000.4750.4830.4560.3090.3610.3490.2370.191-0.214 1.5-1.60.4320.5400.5130.5490.2590.5490.5710.5780.4751.0000.9700.9690.7910.9410.4210.3000.255-0.204 1.6-1.70.3730.4640.4470.4770.2300.5380.5660.5830.4830.9701.0000.9980.8010.9720.5540.4090.359-0.237 1.7-1.80.3560.4260.4010.4290.2140.5070.5320.5490.4560.9690.9981.0000.8040.9820.5370.3990.351-0.231 1.8-1.90.6610.5790.3080.3500.2080.3500.3350.3460.3090.7910.8010.8041.0000.8680.4100.3120.280-0.147 1.9-2.00.3550.3670.2870.3120.1680.3930.4070.4260.3610.9410.9720.9820.8681.0000.5040.3820.341-0.195 2.0-2.20.1500.2240.2480.2630.1180.3560.3890.4180.3490.4210.5540.5370.4100.5041.0000.9400.399-0.204 2.2-2.40.0890.1300.1430.1520.0690.2340.2580.2810.2370.3000.4090.3990.3120.3820.9401.0000.310-0.148 2.4-2.60.0600.0890.0990.1050.0470.1810.2030.2250.1910.2550.3590.3510.2800.3410.3990.3101.0000.099 2.6-2.8-0.166-0.162-0.148-0.155-0.104-0.253-0.257-0.260-0.214-0.204-0.237-0.231-0.147-0.195-0.204-0.1480.0991.000
Table11.7:Correlationcoefficientforthetotalerrorsbetweenmassbins. mXs0.6-0.70.7-0.80.8-0.90.9-1.01.0-1.11.1-1.21.2-1.31.3-1.41.4-1.51.5-1.61.6-1.71.7-1.81.8-1.91.9-2.02.0-2.22.2-2.42.4-2.62.6-2.8 0.6-0.71.0000.0250.0550.0560.0380.0450.0560.0540.0480.0460.0400.0390.0760.0360.0170.0090.006-0.014 0.7-0.80.0251.0000.1820.1720.0830.1020.1410.1370.1170.1080.0940.0890.1250.0700.0470.0260.018-0.026 0.8-0.90.0550.1821.0000.6970.2830.3780.5560.5440.4510.3870.3420.3140.2510.2070.1980.1070.074-0.091 0.9-1.00.0560.1720.6971.0000.2680.3760.5520.5400.4490.3880.3420.3140.2670.2100.1960.1060.073-0.089 1.0-1.10.0380.0830.2830.2681.0000.1920.2650.2570.2170.1830.1650.1570.1590.1140.0880.0480.033-0.060 1.1-1.20.0450.1020.3780.3760.1921.0000.5520.5460.5510.2940.2930.2820.2030.2010.2020.1240.096-0.111 1.2-1.30.0560.1410.5560.5520.2650.5521.0000.7620.7530.4230.4250.4090.2680.2880.3050.1890.148-0.155 1.3-1.40.0540.1370.5440.5400.2570.5460.7621.0000.7510.4250.4360.4190.2760.3000.3250.2050.164-0.156 1.4-1.50.0480.1170.4510.4490.2170.5510.7530.7511.0000.3610.3730.3600.2550.2630.2810.1780.144-0.133 1.5-1.60.0460.1080.3870.3880.1830.2940.4230.4250.3611.0000.7030.7180.6110.6430.3170.2120.180-0.119 1.6-1.70.0400.0940.3420.3420.1650.2930.4250.4360.3730.7031.0000.7510.6290.6740.4240.2930.257-0.140 1.7-1.80.0390.0890.3140.3140.1570.2820.4090.4190.3600.7180.7511.0000.6450.6960.4210.2920.257-0.139 1.8-1.90.0760.1250.2510.2670.1590.2030.2680.2760.2550.6110.6290.6451.0000.6420.3350.2380.214-0.092 1.9-2.00.0360.0700.2070.2100.1140.2010.2880.3000.2630.6430.6740.6960.6421.0000.3640.2580.230-0.109 2.0-2.20.0170.0470.1980.1960.0880.2020.3050.3250.2810.3170.4240.4210.3350.3641.0000.7010.297-0.126 2.2-2.40.0090.0260.1070.1060.0480.1240.1890.2050.1780.2120.2930.2920.2380.2580.7011.0000.216-0.085 2.4-2.60.0060.0180.0740.0730.0330.0960.1480.1640.1440.1800.2570.2570.2140.2300.2970.2161.0000.057 2.6-2.8-0.014-0.026-0.091-0.089-0.060-0.111-0.155-0.156-0.133-0.119-0.140-0.139-0.092-0.109-0.126-0.0850.0571.000
where BFi is the PBF found in bin iwhen evaluating these in bins of photon energy, and Eγiis the photon energy corresponding to the center of the bin (mean photon energy within the bin). We report the mean (M1) and variance (M2−M12) calculated at multiple photon energy cutoffs in Table 11.8.
Table 11.8: The moments of the photon energy spectrum, calculated at multiple photon energy cutoffs. The errors are statistical and systematic, calculated as described in the text.
Eγmin hEi
E2
− hEi2 Preliminary Results
1.897 2.346±0.018+0.027−0.022 0.0211±0.0057+0.0055−0.0069 1.999 2.338±0.010+0.020−0.017 0.0239±0.0018+0.0023−0.0030 2.094 2.365±0.006+0.016−0.010 0.0176±0.0009+0.0009−0.0016 2.181 2.391±0.003+0.008−0.007 0.0129±0.0003+0.0005−0.0005 2.261 2.427±0.002+0.006−0.006 0.0082±0.0002+0.0002−0.0002
Previous Sum of Exclusives Results 1.897 2.321±0.038+0.017−0.038 0.0253±0.0101+0.0041−0.0028 1.999 2.314±0.023+0.014−0.029 0.0273±0.0037+0.0015−0.0015 2.094 2.357±0.017+0.007−0.017 0.0183±0.0023+0.0010−0.0007 2.181 2.396±0.013+0.003−0.009 0.0115±0.0014+0.0005−0.0003 2.261 2.425±0.009+0.002−0.004 0.0075±0.0007+0.0002−0.0002
Since the quantities themselves are not linear relations between the measured PBFs, the uncertainties on these quantities (also reported in Table 11.8) cannot be assumed to be Gaussian distributed, and we therefore simulate the uncertainties, rather than turn to more naive error propagation methods. When doing this, we also need to account for the correlations between bins in our systematic errors (though the different systematic errors themselves, given in Table 11.2, are uncorrelated with one another). One way to do this would be to input an 18-dimensional Gaussian PDF with the correlation matrix given in Table 11.7 into a program that could use it to generate values for our 18 PBFs (distributed according to this Gaussian PDF), and calculate the resulting moment a large number of times. We have not come up with a way, however, to successfully do this exact solution since such programs generally normalize the PDF, or ensure a normalized PDF, which requires a numeric integral over 18 dimensions, and is not feasable.
Our work around is to consider the uncorrelated uncertaintiess separately (there are eight
of these, seven systematic and one statistical), then combine the resulting uncertainties on the mean and variance in quadrature.
For the statistical error on each PBF, uncorrelated between the mass bins, we assume the 18 measured PBFs to be Gaussian distributed and take the measured values as the central values for each of the 18 Gaussians, and the statistical uncertainties as the widths.
We then generate 18 BF values, one for each bin, and re-evaluate the mean and variance at each of our photon energy cutoff values. We repeat this procedure, generating 18 points and calculating the mean and variance, 105 times to obtain distributions of these quantities.
We take as our uncertainty on the calculated value the difference between the central point (which we take to be the peak location of the calculated quantity, slightly different in general than the mean due to the asymmetry of the distribution) and the points that correspond to 16% of the integral of the curve (so the generally asymmetric 68% coverage region). For variance distributions that have a negative tail (an unphysical variance), we only consider
> 0 values for determining our 16% coverage regions. The distributions of the mean and variance for the lowest photon energy cutoff are shown in Figure 11.8.
<E>
2.3 2.32 2.34 2.36 2.38 2.4 2.42 2.44
nStudies/0.000170945506
0 50 100 150 200 250 300 350 400 Mean Distribution
>-<E>2
<E2
-0.02 -0.01 0 0.01 0.02 0.03
nStudies/6.4457511e-05
0 100 200 300 400 500 Mean Distribution
Figure 11.8: The mean (left) and variance (right) distributions used in calculating the error on these quantities. The vertical lines reflect the 16% integrals (68% coverage in the central region).
For the uncertainties that are correlated between bins, we introduce this correlation
“by hand.” We again interpret the uncertainties to reflect Gaussian distributions about the central, measured value, and produce 18 Gaussian PDFs with widths corresponding to the uncertainties reported in Table 11.2. However, for these correlated uncertainties, we produce a 19th Gaussian, a “reference Gaussian,” which is centered on 0 and has a width of
1. We use this reference Gaussian to dictate how much of a shift we should impose on our 18 PDFs simultaneously (we take the value of the PBF in a given mass bin corresponding to a shift by the number of sigma dictated by the reference Gaussian). This way we are recovering the completely correlated error behaviour (a 1-σ shift up in one mass bin is simultaneous with a 1-σ shift up in another) in reasonable computation time.
For uncertainties which have groups of bins that are completely correlated (such as the fragmentation uncertainties), but these groups are not correlated with one another, we simply use multiple reference Gaussians (one for each correlated group of bins).
Propagating the uncertainty in this manner also allows us to evaluate the correlations in the uncertainties between different determinations of the calculated values (between mean and variance, or mean evaluated at different photon energy cutoffs, etc., an example can be seen in Figure 11.9). Each of the uncertainties produces a covariance matrix for these correlations; we add the covariance matrices (and divide by the total error) to get the total correlation coefficients. We report the correlation coefficients for the statistics errors in Table 11.9, for all of the systematic errors in Table 11.10, and for the combined statistics and systematics correlation in Table 11.11. We want to stress, however, that although we present these values as being independent of one another, there are only six independent quantities between the 10 means and variances we report (once six of the quantities are determined, the remaining four are uniquely defined). Nevertheless, we present all of the values in the table since they may be of interest to different calculations.