Averaging of the inelastic cross-section measured by the CDF and the E811 experiments.

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1 Averagg of the astic cross-section measured by the CDF and the E8 experiments. S. Klimenko, J. Konigsberg, T. Liss. Introduction In un II the Tevatron lumosity is measured usg the system of Cherenkov Lumosity Counters (CLC) [,]. It measures the rate of the process of astic pp scatterg, which is proportional to the lumosity = L εclc, where is the astic p p cross-section, ε CLC is the CLC acceptance and L is the lumosity. At the center of mass energy.8 TeV the pp scatterg cross-sections (total, astic and astic) were measured by the E70 [3], CDF [4] and E8 [5] experiments.. Total and astic cross-sections Both experiments measured the total cross-section usg the lumosity dependent method = b N tot 6π ( hc), + ρ N + N where N is the rate of astic and N is the rate of astic p p scatterg events. The slope b is defed as dn b = t 0 N dt and it is exactly the same for both experiments (see Table ). For the slope the E8 is usg the average of the CDF and the E70 measurements with the error domated by the CDF measurement. Therefore comparg the CDF and E8 measurements, the b uncertaty should be excluded. The astic cross-section is b N N = 6π ( hc). + ρ ( N + N ) Introducg the ratio of the astic and astic rates, we can re-write the astic cross-section as b = 6π ( hc). + ρ ( + ) 3. Measured values In both experiments the measured values are the number of astic and astic events. Table shows the slope b, the number of astic and astic events and their ratio. CDF E8 N 7869 ± K ± 3.5K N 4098 ± K ± 57.K

2 3.06 ± ± 0.3 b 6.98 ± ± 0. Table. Input numbers for the astic cross-section. For comparison of two experiments we should ignore the uncertaty of the slope b and compare the measured values of only. The values of and therefore all derived crosssections disagree with the 3.6 standard deviation discrepancy. To localize the source of disagreement lets look at how the astic rates were measured. The astic rate was measured as a sum of the double-arm rate N (cocidence of two detectors measurg astic rates the p and p directions) and the sgle-arm rate N. We could normalize all the rates by N : N N x =, y =, = x + y. N N The measurement of the N and N rates was similar both cases and the N measurement was the most straightforward one. The measurement of the rate N was quite different. The CDF estimated the rate of the sgle diffractive (SD) events by measurg the cocidence rate of the p astic detector with the opposite astic detector (we will call it the sgle diffractive rate ). After applyg the section cuts this rate had a little background, however it required a considerable acceptance and detection efficiency corrections to obta the sgle diffractive rate. To avoid double countg, the double-arm sgle diffractive events were subtracted from the total number of the sgle diffractive events, which gives the estimation of the N rate (309±503 events). The small contribution of the sgle-arm events from the non-diffractive processes (0.6%) was added to the N rate as a simulation-calculated correction. Therefore the corrected number of the sgle-arm events is 33403±50. The non-diffractive correction could be mod dependent. We could image some astic sub-process (for example, we could call it double diffractive ), which gives small contribution both to the double-arm astic rate and the sgle diffractive rate, but a considerable contribution to the sgle-arm rate. If such a sub-process exists, it may not be cluded to the astic rate measured by the CDF. However, it s not liky that the CDF has missed a considerable fraction of the astic events. First, the CDF measurement agrees with UA4 experiment at low energies. Second, accordg to Paolo, the CDF made a cross-check of the sgle diffractive events by measurg the sgle-arm rate. The E8 measured the exclusive sgle-arm rate usg the astic detectors, which should be quite efficient for the sgle diffractive events. However the background from losses was large (~93%). To obta the 3% error quoted on the number of sgle-arm astic events, it required the measurement of the background with uncertaty better then %, which is a non-trivial task. The measurement of the sgle-arm rate was done durg a special run with missg bunches. Therefore order to use it the analysis, fact, the ratio of the sgle-arm and double-arm rates was measured: r = 0.30 ±

3 A small correction to this number due to the fal acceptance is δ=0.007± The total number of the sgle-arm events was estimated as N = N ( r + δ ). Table shows the x and y values measured by the CDF and E8 experiments CDF E8 X.638 ± ± 0.03 Y 0.44 ± ± 0.5 Table. The x and y ratios measured by the CDF and the E8. The x values are a very good agreement, but the y values disagree, which was poted out by Paolo as the source of the CDF-E8 disagreement. It was terpreted as that the E8 has by factor of two more sgle diffractive events, possibly due to the error of the large background subtraction. However, we can t do the direct comparison of the x and y values, because they have different expectation values. The CDF and E8 astic detectors had very different acceptances for the two-side events: ε ( CDF) 98. 7%, ε ( E8) = ±.0%. The E8 sgle-arm rate had a lot of non-diffractive events missed by the two-side astic trigger and the CDF N rate was due to the sgle diffractive process only. Therefore order to check if the E8 astic rates are consistent with the CDF rates we need to take to account the acceptance corrections. To make more tligent comparison we need to estimate the non-diffractive and diffractive (e.g. SD) rates for both experiments. It s straightforward for the CDF and for the E8 the rates are ε N nd = N / ε, N sd = N ( r + δ ). ε There should be a few percent correction for the N sd (E8) rate to account for the doublearm SD events, which we ignore at this moment. Table 3 shows the rates and their ratios to the astic rate. CDF E8 N nd 0300 ± K ± 34.9K N sd 3778 ± K ± 36.3K N nd /N.58± ±0.069 N sd /N 0.480± ±0.07 N sd /N nd 0.86± ±0.04 Table 3. The x and y ratios measured by the CDF and the E8. At this time we have a remarkable agreement between the CDF and the E8 for the ratio N sd /N nd. So both experiments see the same fraction of the sgle diffractive events. At the same time there is the discrepancy of 4.4 standard deviations between the ratios of the non-diffractive astic and astic events. Similar ratio for the sgle diffractive rate is also greater for the E8, but the errors are large and the SD ratios are compatible. So, it is possible, the source of the CDF/E8 discrepancy is the measurement of the astic rates. Unfortunaty, no detail documentation was found on the measurement of the The numbers are shown exactly as they appear the article and C.Avila s talk.

4 astic rate by the E8 experiment. Unless the documentation is provided and we fd maybe more tligent way, there are the followg methods of averagg of the CDF and the E8 measurements. 4. Averagg of the CDF and E8 measurements. Method A. To fd the mean value of the astic cross-section we should average the measurements, which are not compatible. The PDG suggests the followg algorithm: Fd the average of two experiments usg the standard approach: = Fd the average error usg the standard approach: = Calculate χ : 3. Scale the error to get χ = : = 0.. At the first approximation, ignorg the corration of the slope b and, the astic cross-section rative error is = b δ + = (3.8%). b + Fally the average astic cross-section is ( + ρ ) = 60.4 ±.3 mb, which is.% bow the CDF measurement. Method B. Now lets take to account the corration between the slope b and the ratios. As Heidi mentioned, the astic rate is the raw n rate measured each experiment divided by its acceptance N = n / exp bt exp bt, ( ( ) ( )) m where ( tm, tmax ) is a range of t used by the CDF ( < t < 0. 9 ) and E8 ( < t < ). Both measurements depend on the slope b and, fact, they are anti-corrated. Namy, if we crease b by one standard deviation (.5%), the CDF value of creases by ~% and the E8 value decreases by ~%. The covariance matrix cov(i, j ) is α C =, α where ( ) is the standard deviation of the ratio ( ) for the CDF (E8) measurement and the coefficient α is estimated to be The average value of the ratio is = f + ( f ), where the weight f max For details see Heidi s note.

5 α f = + α can be found by mimization of the variance of T var( ) = FCF, F = ( f, f ). The same result can be obtaed by mimization of the likihood L = log( p(,,,, ), where p is the probability distribution function of two corrated Gaussian variables ( ( ( ( p = exp α +. π ( ) α α Assumg that the standard deviations, are given by the errors listed Table 3, the average value of is = 3.0 ± 0.06, which is very close to the number obtaed by the method A. Calculatg the χ ( i χ = C i, j i ij ( j ;( i, j =,) and applyg the same procedure for the scalg of the error of, the astic crosssection is ( + ρ ) = 60.3 ±. mb. j Method C. The averagg procedure described above can be applied to the astic and astic cross-sections measured by the CDF and E8. Usg the functional dependences of the astic and astic cross-sections on the slope b and the ratio, lets, first, derive the mean value and the error for each experiment and compare with quoted numbers (see Table 4). CDF E8 Quoted tot, mb ± ±.0 Derived tot, mb ± ±.90 Quoted, mb ± ±.9 Derived, mb 60.3 ± ±.5 Table 4. Comparison of the values of the cross-sections and their errors quoted by the CDF ( ρ = 0.5) and E8 ( ρ = 0.45) and derived this note. The derived errors are slightly smaller, however they all are compatible with the quoted errors. It means that the errors are maly determed by the errors of the slope b and the ratio. The averaged cross-sections with their flated errors are tot ( + ρ ) = 76.8 ± 4.7 mb, 3 Actually the E8 error is underestimated, because the error of the slope b is ignored the ratio.

6 ( + ρ ) = 58.8 ±.7 mb. 5. Comparison of different methods. Table 5 shows the average cross-sections for all three methods. ( + ρ ) + ρ tot ( ) Method A 60.4 ± ± 4. Method B 60.3 ±. 79. ± 4.0 Method C 58.8 ± ± 4.7 Table 5. Average cross-sections. The methods A and B are based on the averagg of the ratio. Sce the measurements disagree for the CDF and E8, the error of the average value of should be flated. In this way we ignore the accurate error analysis done by both experiments and, order to calculate the fal cross-sections, we don t care about the corration of the errors of b and anymore. The method B is a more accurate version of the method A and gives almost the same result. With the method C we average the quoted cross-sections itsf. It seems to be less straightforward then the methods A and B. First, one needs to estimate the covariance CDF E8 CDF E8 cov(, ) 0.4, cov( tot, ) 0.4. tot, ) 0. They are large compare to the covariance cov( 09, which can be neglected as the comparison of the methods A and B shows. Second, it does not show the actual lev of disagreement between the CDF and E8 measurements. For example, the χ values calculated for different measurements are χ =.0, χtot = 8.6, χ = 6. 6 and the measurements agree with the confidence lev of CL < 0.%, CL tot 0.3%, CL.0% respectivy. In fact, we can t even say that there is a disagreement for the astic crosssection. Therefore we suggest the method A for averagg of the CDF and E8 measurements. Usg the ρ value of 0.35 the average astic cross-section at s = 800 GeV is = 59.3 ± Inastic cross-section s = 960 GeV. In un-ii the Tevatron center of mass energy is.96 TeV. There are no measurements of the p p scatterg cross-sections at this energy. Therefore the extrapolated value of the astic cross-section is used for the lumosity measurements. Theory predicts [7,8] that the astic cross-section creases with energy as ln s and the diffraction cross-section creases as ln s. The total cross-section is a mixture of the astic and astic processes, where the astic cross-section creases more rapidly with energy. The energy dependence of obtaed from the best fit of the experimental data (cludg cosmic tot. rays data) is ln s [9], which is consistent with the expectations. However the E70 and

7 E8 measurements favor the ln s dependence. Table 6 shows the value of the astic cross-section at.96 TeV and it s rative variation for all three options. (.96TeV) δ /,% Ln(s) 60.0 ±.3. ln (s) 60.7 ±.3.3 ln. (s) 60.8 ±.3.6 Table 6. Extrapolated astic cross-section at.96 GeV. Assumg the ln s energy dependence of the astic cross-section and assigng an additional % systematic error due to uncertaty the energy dependence the astic cross-section at.96 TeV is (.96TeV ) = 60.7 ± How the CDF lumosity is affected. The CDF lumosity is derived from the rate of the astic p p events measured with the lumosity monitor (CLC), the CLC acceptance and the astic cross-section L = εclc with the systematic errors of.8%, 4.0% and 3.8% respectivy. The total systematic error on the lumosity is 5.8%. Also the new value of the astic cross-section will shift the mean value of the lumosity. The astic processes can be divided on three groups: a) hard-core, b) sgle diffractive, c) double diffractive. The sgle diffractive cross-section was measured by the CDF and E70 experiments: 9.46 ± mb (CDF), 8. ±. 7 mb,.7 ±.3mb (E70). Sce the average sgle diffractive cross-section is not much different from the CDF measurement, I will use the CDF number. The double diffractive cross-section was measured by CDF (PL 87, 00) and accordg to Mary Convery it is 7 ± mb. The rest of the astic cross-section ( = ), where the astic crosssection is given for ρ=0.35, is the hard-core cross-section. All the numbers are given for the center of mass energy of.8 TeV. At.96 TeV the CLC group used the followg values of the astic cross-sections from the MB generator: = mb, = 0. 3, = Therefore the total hc astic cross-section of = 6. 7 mb at.96 TeV was used for the lumosity estimation. This value is obtaed from the CDF measurement at.8 TeV and extrapolated to.96 TeV usg the ln s energy dependence. The CLC acceptance was estimated to be 60.±.4% assumg the rative fraction of the astic processes 44.4/0.3/7.0. The CLC acceptances for each process were estimated to be ε hc = 79.%, ε sd = 9.0%, ε dd = 6.5%. Given the rative fraction of 43.95/9.46/7.0 at.8 TeV, the CLC acceptance is 60.8%, which is very close to the number above. If the Tevatron energy would be.8 TeV, the lumosity correction is pp sd dd

8 δl ( CDF) 60.4 = = = L 59.3 Assumg that the ln s extrapolation for the astic cross-section from.8tev to.96tev, the same correction of +.9% should be applied to the CLC lumosity at.96tev. 8. Conclusion 9. eferences. D. Acosta et al. Nucl. Instr. and Meth. A46 (00) 540. D. Acosta et al. Nucl. Instr. and Meth. A494 (00) N.A. Amos et al. Phys.ev.Let, 68, p433, F. Abe et al. Phys.ev.D, 50, p5550, N.A. Amos et al. Phys.Let.B, 445, p49, F. Abe et al. Phys.ev.D, 50, p5535, A. Capla et al. Zeit. Phys. C3 (980) A. Capla et al. Phys. eports 36 (994) 5 9. C. Augier et al. Phys. Lett. B35 (993) 503