1. In the context of the HERA-LHC workshop the idea of combining the H1 and ZEUS data arose. Not just putting both data sets into a common PDF fit but actually averaging the data first. Why? Well, because combining ZEUS and H1 data in fits has not been very successful, e.g. MRST say ‘ZEUS and H1 data sets pull against each other’ → let’s not rely on MRST, so I tried looking at H1 data myself, with the ZEUS PDF analysis I also collaborated with Sasha Glazov to ensure that the averaging of ZEUS and H1 was done with the correct ZEUS data sets and correlations (and this enabled me to understand the H1 correlations!) I then made a PDF fit to the resulting combined HERA data and compared it to fitting the data sets separately And I made a brief model dependence analysis of this fit Combining ZEUS and H1 data sets? A M Cooper-Sarkar Oct 06 Let’s refresh our memories- all HERA-I in this talk, and NO jets.

2. ZEUS analysis/ZEUS data ZEUS analysis/H1 data ZEUS analysis/H1 data compared to H1 analysis/H1 data Here we see the effect of differences in the data, recall that the gluon is not directly measured (no jets) The data differences are most notable in the large 96/97 NC samples at low-Q2 The data are NOT incompatible, but seem to ‘pull against each other’ IF a fit is done to ZEUS and H1 together the χ2 for both these data sets rise compared to when they are fitted separately……….. Here we see the effect of differences of analysis choice - form of parametrization at Q2_0 etc

3. See if you can spot the data differences between ZEUS/H1 at low Q2..It is mostly in slope.

4. ZEUS ONLYZEUS+H1 Look at the effect of ADDING H1 data to the ZEUS data in the ZEUS PDF analysis- the OFFSET method is used for the correlated systematic errors Adding H1 data does NOT significantly improve errors on the gluon - statistical uncertainty improves - but systematic uncertainty does not -χ2 for each data set increases- and OFFSET errors reflect this

5. ZEUS ONLYZEUS+H1 Whereas adding H1 to ZEUS data brings no big improvement for the sea and gluon determination, it does bring improvement to the high-x valence distributions, where statistical errors dominate The ZEUS and H1 high-Q2 data are also seem more compatible – there must be an advantage in having a joint H1/ZEUS data set?

6. So it is hoped that combining the data sets could bring real advantages in decreasing the PDF errors, if the low-Q 2 discrepancies in the data sets can be resolved. How could this be done? Any combination of the data points would have to be done accounting for the correlated systematic errors One could use the HESSIAN method Essentially the Hessian method of combination can swim each experiment towards the other within the tolerance of the systematic errors of each data set. Let’s remind ourselves what the Hessian method and Offset methods are in PDF fitting……

7. Experimental systematic errors are correlated between data points, so the correct form of the χ2 is χ 2 = Σ i Σ j [ F i QCD (p) – F i MEAS ] V ij - 1 [ F j QCD (p) – F j MEAS ] V ij = δ ij (б i STAT ) 2 + Σ λ Δ iλ SYS Δ jλ SYS Where ) i 8 SYS is the correlated error on point i due to systematic error source λ It can be established that this is equivalent to χ 2 = 3 i [ F i QCD (p) – 3 8 s  i SYS – F i MEAS ] 2 + 3 s 2 (  i STAT ) 2 Where s 8 are systematic uncertainty fit parameters of zero mean and unit variance This form modifies the fit prediction by each source of systematic uncertainty

8. How ZEUS uses this: OFFSET method 1.Perform fit without correlated errors (s λ = 0) for central fit 2.Shift measurement to upper limit of one of its systematic uncertainties (s λ = +1) 3.Redo fit, record differences of parameters from those of step 1 4.Go back to 2, shift measurement to lower limit (s λ = -1) 5.Go back to 2, repeat 2-4 for next source of systematic uncertainty 6.Add all deviations from central fit in quadrature (positive and negative deviations added in quadrature separately) 7.This method does not assume that correlated systematic uncertainties are Gaussian distributed Of course it is actually done in a very smart quick way using matrices, follwing Botje

9. HESSIAN method Allow s λ parameters to vary for the central fit 1.The total covariance matrix is then the inverse of a single Hessian matrix expressing the variation of χ2 wrt both theoretical and systematic uncertainty parameters. 2.If we believe the theory why not let it calibrate the detector(s)? Effectively the theoretical prediction is not fitted to the central values of published experimental data, but allows these data points to move collectively according to their correlated systematic uncertainties 3.The fit determines the optimal settings for correlated systematic shifts s λ such that the most consistent fit to all data sets is obtained. In a global fit the systematic uncertainties of one experiment will correlate to those of another through the fit 4.We must be very confident of the theory to trust it for calibration– but more dubiously we must be very confident of the model choices we made in setting boundary conditions to the theory 5.CTEQ use this method but then raise the χ2 tolerance to Δχ2=100 to account for inconsistencies between data sets and model uncertainties. H1 use it on their own data only with Δχ2=1

10. The HESSIAN method does give a smaller estimated of the PDF errors than the OFFSET method if you stick to Δχ2=1 Comparison off HESSIAN and OFFSET methods for ZEUS-JETS fit However it gives larger model errors, because each change of model assumption can give a different set of systematic error parameters, and thus a different estimate of the shifted positions of the data points. Compare the latest H1 and ZEUS PDFs –SEE next slide—in the end there is no great advantage in the Hessian method.. For the gluon and sea distributions the Hessian method gives a much narrower error band. Equivalent to raising the Δχ2 in the Offset method to 50.

11. Compare the latest H1 and ZEUS PDFs –in the end there is no great advantage in the Hessian method for the ZEUS- ONLy or H1-ONLY PDF fits, because the model dependence ‘cancels it out’. But the current proposal is not to do a Hessian PDF fit of both data sets BUT to do a Hessian fit to combine the data points without model assumptions..

12. ASIDE: I Have always distrusted the Hessian method – WHY? Because the fitted values of the systematic error parameters change A LOT according to model assumptions and according to the different data sets used in the fit. e.g. one is using the fit to calibrate the data, and one PDF analysis tells us that the correct setting for our RCAL energy scale is up by 2.4%, whereas another PDF analysis tells us that it is down by 1.8% - they can’t both be true- compare CTEQ and ZEUS-S global fit analyses (both done by Hessian method) for ZEUS systematic error parameters for the NC 96/97 data set Zeus sλCTEQ6ZEUS-S 1 1.67-0.36 2 -0.671.17 3 -1.251.20 4 -0.440.40 5 0.000.32 6 -1.070.39 7 1.28-1.40 8 0.620.20 9 -0.400.04 10 0.21-0.06

13. So to return to the idea of combining or averaging the ZEUS/H1 data accounting for the correlated systematic errors …. Using the Hessian method………. without any theoretical model. What does this mean? Briefly For each cross-section measurement (NC e+/e- CCe+/e-), each x,Q 2 point must have a true value of the cross-section Let this true value be a parameter of a new ZEUS+H1 averaging fit, ie there is one parameter for each x,Q 2 point We now have at least two measurements –one from ZEUS one from H1 to determine this parameter ( sometimes, eg for e+ p, there maybe more than one data set per experiment –e+ p 96/97 and e+ p 99/00 ) The systematic error parameters, s λ, for each experiment, must also be parameters of this fit and all of the x,Q 2 points for all of the data sets are fitted at once so that correlated systematic errors between data points, and between data sets, are included. E.G For NC e+p 96/97 there are ~250 x,Q2 points per expt. So there will be 250 parameters for the true values of the cross section and ~20 systematic parameters (~10 per expt.). These are fitted to the ~500 total data points.

14. Essentially the Hessian method of combination swims each experiment towards the other within the tolerance of the systematic errors of each data set. Technical matters: 1.Have agreed the exact ZEUS and H1 data sets to be used with S. Glazov of H1 2.Have agreed treatment of correlations within each experiment 3.H1/ ZEUS measurements are not in fact at the same x,Q 2 values so need to agree x,Q 2 grid: so used an H1 optimized grid and checked with a ZEUS optimized grid 4.Points which are Not on this grid must be swum to it, used H1’s fractal fit to do the swimming. However results are not sensitive to the grid choice and hence not sensitive to the swimming procedure SO what does it look like? It has 1153 data points: ZEUS+H1 548 free parameters for all the x,Q 2 points of the grid for the 4 xsecns (NC/CC e+/e-) Plus 18 (H1) + 26 (ZEUS) independent systematic parameters, s λ Χ 2 =579 for (1153-548-44)=561 degrees of freedom

15. Averaging does not favour one expt or the other Q2=500Q2=1000 CC e- Red is H1 Green is ZEUS Black is the HERA average – slightly displaced so you can see the size of its error compared to the input ZEUS/H1 points CC e+ Red is H1 94/97 Green is H1 99/00 Blue is ZEUS 94/97 Yellow is ZEUS 99/00 Black is the HERA average

16. Averaging does not favour one expt or the other Q2=250 Q2=2000 Q2=90 Q2=650 NC e- Red is H1 Green is H1 Fl Blue is ZEUS Black is the HERAaverage– slightly displaced so you can see the size of its error compared to the input ZEUS/H1 points NC e+ Red is H1 bulk 96/97 Green is H1 mb 96/97 Blue is H1 highq2 96/97 Yellow is H1 99/00 Diamonds are ZEUS 96/97 Squares are ZEUS 96/97 Triangles are ZEUS 99/00 Full Black is the HERA average

17. Averaging does favour H1 to some extent at Q2~ 15 - 65 Q2=3.5 Q2=6.5 NC e+ at lower Q2 Red is H1 bulk 96/97 Green is H1 mb 96/97 Blue is H1 highq2 96/97 Yellow is H1 99/00 Diamonds are ZEUS 96/97 Squares are ZEUS 96/97 Triangles are ZEUS 99/00 Full Black is the HERA average – slightly displaced so you can see the size of its error compared to the input ZEUS/H1 points Q2=15 Q2=25

18. NC96/97 systematic S λ, λ=1,10 ZEUSgrid H1grid 1 zd1_e_eff 0.3 0.15 2 zd2_e_theta_a -0.07 0.4 3 zd3_e_theta_b -0.45 -0.21 4 zd4_e_escale 0.97 1.04 5 zd5_had1 0.36 0.31 6 zd6_had2 0.4 0.39 7 zd7_had3 -1.1 -0.62 8 zd8_had_flow 0.05 0.84 9 zd9_bg -0.07 -0.38 10zd10_had_flow_b -0.06 -0.26 Technical checks Here are the NC 96/97 ZEUS systematic error parameters as determined by the averaging fit, using the two different x,Q2 grids Reasonable consistency between choices of grid- see EXTRAS for comparison of H1, ZEUS and combined points using the ZEUS grid s λ shifts are not so big (see also next slide) – remember s λ = ±1`represents a one б shift

19. CC e+CC e- NC e- NC e+ No of standard devns the original data points are pulled by the fit Vs Q 2 for each data set

20. This means s λ This means the uncertainty on s λ, Δs λ i.e Δs λ, becoming smaller

21. 1 zlumi1_zncepl -1.2841 0.5836 20 zd1_e_eff 0.0372 0.7500 21 zd2_e_theta_a 0.4317 0.6674 22 zd3_e_theta_b -0.1718 0.7836 23 zd4_e_escale 1.0641 0.4874 24 zd5_had1 0.3632 0.5873 25 zd6_had2 0.3600 0.6440 26 zd7_had3 -0.6905 0.7383 27 zd8_had_flow 0.8052 0.6519 28 zd9_bg -0.2694 0.4054 29 zd10_had_flow_b -0.2623 0.1686 30 zd11 -0.6408 0.5995 31 z1nce-_e_scale 0.1555 0.9384 32 z2nce-_bg -0.3000 0.9215 33 z3nce-_eff1 0.0076 0.9310 34 z4nce-_eff2 0.3770 0.5543 35 z5nce-_vtx -0.6828 0.9274 36 z6nce- 0.2241 0.4158 37 z1cce- 0.1846 0.8409 38 z2cce- 0.1920 0.8687 39 zlumi2_zccem -0.7516 0.7739 40 zd5nc00 0.0835 0.9842 41 zd7nc00 0.1174 0.2068 42 zd8nc00 0.3087 0.9258 43 zluminc -1.8738 0.3755 Let’s look at ZEUS instead Large reductions in uncertainty Δs λ are highlighted What we NEED is a large reduction in any systematic uncertainty which is a big contributor to the total uncertainty For example, there is an impressive reduction in uncertainty in the photo- production background for the 96-97 NC data How does this come about….see next slide Systematic s λ Δs λ Large shifts i.e. large values of s λ are also highlighted- these are mostly in normalisations And most of the normalisation shift is obtained by averaging ZEUS to itself (e.g. 96/97 to 99/00) rather than to H1! - See EXTRAS

22. H1 ZEUS The fit calibrates one experiment against the other

23. Let’s see some results –HERA averaged data points for low Q 2 NC e+ Q2 x б Δб(stat) Δб(sys) Δб(tot) 2.5000 0.0001 0.7426 0.0102 0.0073 0.0126 2.5000 0.0002 0.7015 0.0195 0.0075 0.0209 2.5000 0.0003 0.6427 0.0145 0.0068 0.0160 2.5000 0.0005 0.6075 0.0137 0.0070 0.0154 2.5000 0.0008 0.5616 0.0162 0.0061 0.0173 2.5000 0.0016 0.5214 0.0140 0.0059 0.0152 2.5000 0.0050 0.4104 0.0161 0.0057 0.0171 3.5000 0.0001 0.8875 0.0130 0.0091 0.0158 3.5000 0.0002 0.8330 0.0122 0.0086 0.0149 3.5000 0.0003 0.7835 0.0122 0.0082 0.0147 3.5000 0.0005 0.7290 0.0117 0.0083 0.0143 3.5000 0.0008 0.6544 0.0126 0.0070 0.0144 3.5000 0.0013 0.6136 0.0119 0.0073 0.0140 3.5000 0.0025 0.5266 0.0062 0.0054 0.0083 3.5000 0.0080 0.4409 0.0076 0.0054 0.0093 5.0000 0.0001 1.0376 0.0151 0.0109 0.0186 5.0000 0.0002 0.9803 0.0143 0.0106 0.0178 5.0000 0.0003 0.9138 0.0145 0.0091 0.0171 5.0000 0.0005 0.8222 0.0125 0.0084 0.0151 5.0000 0.0008 0.7193 0.0135 0.0080 0.0157 5.0000 0.0013 0.6760 0.0130 0.0091 0.0158 5.0000 0.0020 0.6277 0.0123 0.0069 0.0140 5.0000 0.0040 0.5492 0.0092 0.0061 0.0111 5.0000 0.0130 0.4220 0.0137 0.0046 0.0145 Hence a new PDF fit can be done which should be much less sensitive to treatment of systematics… I’ve simply added errors in quadrature Now that all data points have been averaged the systematic errors are smaller than the statistical for all data points- including those at low-Q2, where systematic error was up to 3 times statistical in the separate data sets

24. New PDF fit to the HERA averaged inclusive xsecn data using the ZEUS fit analysis Compare to the published PDF shapes for H1 PDF 2000 and ZEUS-JETS The sea and the u-valence are very similar The χ2 of this 11 parameter quadrature fit are 1.23 for 34 e+ CC data points - errors underestimated? Hence small d valence PDF error 0.59 for 31 e- CC data points – errors overestimated? 1.02 for 318 e+ NC data points 0.81 for 145 e- NC data points The d-valence of the HERA averaged data set is not really like either ZEUS or H1 published d- valence The gluon of the HERA averaged data set is more like the ZEUS published gluon

25. Compare this new PDF fit to the HERA averaged inclusive xsecn data To a PDF fit to H1 and ZEUS inclusive xsecn data NOT averaged –where we get more of a compromise between ZEUS and H1 published PDF shapes The PDF fit to H1 and ZEUS not averaged was done by the ZEUS analysis using the OFFSET method.. We could consider doing it by the HESSIAN method- allowing the systematic errors parameters to be detemined by the fit

26. Compare the PDF fit to the HERA averaged inclusive xsecn data To the PDF fit to H1 and ZEUS inclusive xsecn data NOT averaged –done by the ZEUS PDF analysis but using the HESSIAN method As expected the errors are much more comparable But the central values are rather different This is because the systematic shifts determined by these fits are different

27. systematic shift s λ QCDfit Hessian ZEUS+H1 GLAZOV theory free ZEUS+H1 zd1_e_eff 1.65 0.31 zd2_e_theta_a -0.56 0.38 zd3_e_theta_b -1.26 -0.11 zd4_e_escale -1.04 0.97 zd5_had1 -0.40 0.33 zd6_had2 -0.85 0.39 zd7_had3 1.05 -0.58 zd8_had_flow -0.28 0.83 zd9_bg -0.23 -0.42 zd10_had_flow_b 0.27 -0.26 h2_Ee_Spacal -0.51 0.61 h4_ThetaE_sp -0.19 -0.28 h5_ThetaE_94 0.39 -0.18 h7_H_Scale_S 0.13 0.35 h8_H_Scale_L -0.26 -0.98 h9_Noise_Hca 1.00 -0.63 h10_GP_BG_Sp 0.16 -0.38 h11_GP_BG_LA -0.36 0.97 A very boring slide- but the point is that it may be dangerous to let a QCDfit determine the optimal values for the systematic shift parameters. And using Δχ2=1 on such a fit gives beautiful small PDF uncertainties but a central value which is far from that of the theory free combination!!..

28. Now consider model dependence of PDF fit to HERA averaged data First a reminder of what goes into the standard fit xu v (x) = Au x av (1-x) bu (1 + c u x) xd v (x) = Ad x av (1-x) bd (1 + c d x) xS(x) = As x as (1-x) bs xg(x) = Ag x ag (1-x) bg (1 + c g x) xΔ(x) = AΔ x 0.5 (1-x) bs+2 No sensitivity to shape of Δ= dbar-ubar so AΔ fixed consistent with Gottfried sum-rule Assume sbar = (dbar+ubar)/4 consistent with neutrino dimuon data Au, Ad, Ag are fixed by the number and momentum sum-rules au=ad=av for low-x valence -11 params Call this the STANDARD fit

29. Model variations – see EXTRAS for dbar-ubar variationn 1.(1+c s x) term in the sea param. 2.(1+c g x + e g √x) in the gluon param. 3.xd v (x) = Ad x av (1-x) bd (1 + cd x) + B xu v (x). x(1+x) extra term in d-valence 4.au  ad for low-x valence 5.(1+c u x +e u √x) and (1+c d x +e d √x) for both uv and dv param. These variations are displayed in the next few plots in the order 0. standard 1. sea x term 2. glue √x term 3. dv/uv term 4. low-x valence 5. valence √x terms There is no dramatic change in central values of the PDFs Note the addition of extra freedom does not improve χ2 much Standard χ2 is 504.7 for 529 data points, 11 params 0. Δχ2=0 1. Δχ2= -0.8 2. Δχ2= -10 ( but nonsense- high x gluon becomes negative!) 3. Δχ2=-0.5 4. Δχ2=-0.6 5. Δχ2= -7 ( all from u valence)

32. Conclusions The averaging idea looks promising It is better to combine at the level of the data, than within the PDf fit itself The resultant PDFs look promising Model dependence does not seem severe

33. EXtras Glazov’s own plots of how the HERA average data compares to H1 and ZEUS data points Some plots which show that choice of grid, and hence swimming approximations, do not count for much Slides on averaging ZEUS within itself (i.e. e + 96/97 and 99/00) as compared to averaging ZEUS and H1 Further model dependence variations of the PDF fit to the HERA averaged data

36. Alternative results using ZEUS grid points - Choice of grid is not so crucial Q2=530Q2=950 CC e- Red is H1 Green is ZEUS Black is the average CC e+ Red is H1 94/97 Green is H1 99/00 Blue is ZEUS 94/97 Yellow is ZEUS 99/00 Black is the average

37. Q2=250 Q2=2000 Alternative results using ZEUS grid points - Choice of grid is not so crucial Q2=90 Q2=650 NC e- Red is H1 Green is H1 Fl Blue is ZEUS NC e+ Red is H1 bulk 96/97 Green is H1 mb 96/97 Blue is H1 highq2 96/97 Yellow is H1 99/00 Diamonds are ZEUS 96/97 Squares are ZEUS 96/97 Triangles are ZEUS 99/00

38. Q2=3.5 Q2=6.5 Q2=15 Q2=25 Alternative results using ZEUS grid points - Choice of grid is not so crucial NC e+ at lower Q2 Red is H1 bulk 96/97 Green is H1 mb 96/97 Blue is H1 highq2 96/97 Yellow is H1 99/00 Diamonds are ZEUS 96/97 Squares are ZEUS 96/97 Triangles are ZEUS 99/00

39. 1 zlumi1_zncepl -1.2841 0.5836 20 zd1_e_eff 0.0372 0.7500 21 zd2_e_theta_a 0.4317 0.6674 22 zd3_e_theta_b -0.1718 0.7836 23 zd4_e_escale 1.0641 0.4874 24 zd5_had1 0.3632 0.5873 25 zd6_had2 0.3600 0.6440 26 zd7_had3 -0.6905 0.7383 27 zd8_had_flow 0.8052 0.6519 28 zd9_bg -0.2694 0.4054 29 zd10_had_flow_b -0.2623 0.1686 30 zd11 -0.6408 0.5995 31 z1nce-_e_scale 0.1555 0.9384 32 z2nce-_bg -0.3000 0.9215 33 z3nce-_eff1 0.0076 0.9310 34 z4nce-_eff2 0.3770 0.5543 35 z5nce-_vtx -0.6828 0.9274 36 z6nce- 0.2241 0.4158 37 z1cce- 0.1846 0.8409 38 z2cce- 0.1920 0.8687 39 zlumi2_zccem -0.7516 0.7739 40 zd5nc00 0.0835 0.9842 41 zd7nc00 0.1174 0.2068 42 zd8nc00 0.3087 0.9258 43 zluminc -1.8738 0.3755 1 zlumi1_zncepl -0.6889 0.8305 20 zd1_e_eff -0.0320 0.8939 21 zd2_e_theta_a -0.0104 0.9904 22 zd3_e_theta_b -0.5166 0.8447 23 zd4_e_escale 1.3358 0.6843 24 zd5_had1 0.1296 0.7275 25 zd6_had2 0.3411 0.7531 26 zd7_had3 -0.5524 0.9721 27 zd8_had_flow 0.3981 0.9790 28 zd9_bg 0.0904 0.9970 29 zd10_had_flow_b 0.2050 0.3440 30 zd11 0.0000 1.0000 31 z1nce-_e_scale 0.1372 0.9806 32 z2nce-_bg -0.1003 0.9639 33 z3nce-_eff1 0.0460 0.9739 34 z4nce-_eff2 0.1471 0.6959 35 z5nce-_vtx 0.0000 1.0000 36 z6nce- 0.5106 0.5940 37 z1cce- 0.3715 0.9254 38 z2cce- 0.0837 0.9408 39 zlumi2_zccem 0.0000 1.0000 40 zd5nc00 0.1649 0.9907 41 zd7nc00 0.2065 0.2629 42 zd8nc00 0.0650 0.9547 43 zluminc00 -1.6620 0.6677 ZEUS against H1ZEUS against itself

40. Q2 x б Δб(stat) Δб(sys) Δб(tot) 3.5000 0.0001 1.0764 0.0509 0.1286 0.1383 3.5000 0.0001 0.9660 0.0275 0.0311 0.0415 3.5000 0.0001 0.9014 0.0250 0.0197 0.0318 3.5000 0.0002 0.8483 0.0221 0.0188 0.0290 3.5000 0.0003 0.8094 0.0231 0.0182 0.0295 3.5000 0.0005 0.7291 0.0227 0.0166 0.0281 3.5000 0.0025 0.5247 0.0086 0.0119 0.0146 3.5000 0.0080 0.4680 0.0090 0.0138 0.0165 Q2 x б Δб(stat) Δб(sys) Δб(tot) 3.5000 0.0001 0.9818 0.0384 0.0320 0.0500 3.5000 0.0001 0.9385 0.0158 0.0129 0.0204 3.5000 0.0001 0.8874 0.0130 0.0091 0.0158 3.5000 0.0002 0.8333 0.0122 0.0087 0.0149 3.5000 0.0003 0.7840 0.0122 0.0082 0.0147 3.5000 0.0005 0.7296 0.0117 0.0083 0.0143 3.5000 0.0025 0.5269 0.0062 0.0054 0.0083 3.5000 0.0080 0.4413 0.0076 0.0054 0.0093 ZEUS averaged to itself ZEUS averaged to H1 Averaging with H1 is much better than just averaging ZEUS to itself Not only is there a stronger reduction in statistical error when H1 is added there is a much more significant reduction in systematic error

41. Extra model dependence: dbar – ubar not forced to zero as x → 0 Extra model dependence: (1+c d x +e d √x) in d-valence alone