- Email: [email protected]
Measurements of αs at centre-of-mass energies between 91 and 209 GeV
Nuclear Physics B (Proc. Suppl.) 121 (2003) 59-64
www.clscvier.com/locatclnpc
Measurements of cx, at Centre-of-Mass Energies between 91 and 209 GeV H. StenzeP * aII. Physikalisches Institut, Universitat GieDen, Heinrich-Buff-Ring 16, D-35392 Giei3en, Germany Consistent measurements of event-shape variables and a, using the complete set of data taken with the ALEPH detector at LEP are presented. The strong coupling constant is determined from a fit of 0(a~)+X’LLA QCD calculations to distributions of six event-shape variables at eight centre-of-mass energies. A new method is proposed to estimate theoretical systematic uncertainties. The results are compared to the expected evolution of
1. Introduction
The most important parameter of perturbative QCD is the coupling constant a,. It can be determined in e+e- annihilations at LEP in a wide range of energies, hence testing its expected energy evolution. The ALEPH experiment has collected data at centre-of-mass energies from 91.2 to 209 GeV. The method to extract Q, has to cope with rather different sample sizes at different energies, spanning from a few hundred events at 161 and 172 GeV to a million events at the Z” boson resonance. Under these conditions, distributions of event-shape variables are the most powerful observables. Event-shape distributions have been calculated to second order in os, these calculations are matched to resummed calculations of leading and next-to-leading logarithms to all orders in as. While this technique has been used before and elsewhere, here a new method is presented to determine theoretical uncertainties of the measurement of crs. A thorough investigation of dominant uncertainties related to missing higher orders has been elaborated. 2. Experimental
Data
Interactions above Mz were recorded at various centre-of-mass energies. The data were accumulated at seven nominal energies given in Table 1, some runs in narrow energy ranges were evolved to the closest nominal energy and lumped to*on
behalf
of the
ALEPH
gether. In addition, a sub-sample of high-quality data at LEPI (Q = E,, = 91.2 GeV) containing 1.1 million events was re-analysed. A detailed description of the ALEPH detector is given in [l]. The @(y) events at LEPII (Q 2 133 GeV) are extracted by selecting hadronic events as described in Ref. [2] in a procedure that has several steps. First hadronic events in which a Z is accompanied by an initial state photon radiation (ISR) are removed. Then various cuts on the event topology and jet configuration are applied to reject background from four-fermion processes (WW, ZZ, Zy’). The resulting distributions of event shapes are finally corrected for residual ISR effects. Irreducible background (5-15 %) is subtracted bin-by-bin. Corrections of detector effects and acceptance are applied yielding distributions at hadron level. A definition of the event-shape variables studied here is given in [2]; these are thrust T [3], heavy jet mass squared M&/s [4], wide and total jet broadening Bw and B-J- [5], C-parameter C [6] and - In y3 [7]. A s an example, the resulting distributions of C-parameter are shown in Figure 1. Experimental systematic uncertainties are obtained with variations of cuts, both for event and track selection, and for background rejection and ISR suppression. The dominant experimental uncertainty at LEPII is related to the modeling of corrections for residual ISR, at LEPI it is the simulation of neutral hadronic calorimeter clusters.
collaboration.
0920-5632/03/S - see front matter doi: 10.1016/SO920-5632(03)01813-9
0 2003 Elsevier
Science
B.V.
All rights
reserved
60
II. Stenzel/Nuclear
Physics
B (Proc,
Suppl.)
121 (2003)
59-64
Table 1 Integrated luminosities and numbers of accepted and expected events. There is an uncertainty the predicted numbers of events. JLdt W-l> 12.30 11.08 9.54 58.83 174.36 206.02 216.19
(2) 133 161 172 183 189 200 206
0
0.1
IIIIIIIIIIIIII.I/,IIIIIIIIIIIII 0.2 0.3 0.4 0.5
events found 806 319 257 1319 3578 3514 3578
0.6
0.7
events expected 822 333 242 1262 3578 3528 3590
:. ,-- , , 0.8 0.9 c 1
Figure 1. The measured distributions, after correction for backgrounds and detector effects, of Cparameter at energies between 91.2 and 206 GeV together with the fitted QCD predictions.
3. Measurements
of cxS
The coupling constant oS is determined from a fit of the perturbative QCD prediction to the Measuremeasured event-shape distributions. ments are performed for each variable and each energy first, then different variables are combined into measurements of a,(Q) and finally all data are combined into a single value of a,(Mz).
expected signal 822 319 218 1109 3124 3005 3072
of 2% in
expected background 0.0 14 24 153 454 523 518
Event-shape distributions are fitted in the central region of 3-jet production, where a good perturbative description is available. The fit range is placed inside a region where hadronisation corrections are well under control, where detector corrections are stable and background corrections are small. The perturbative prediction consists of second order calculations 0(az) which are matched to calculations resumming leading and next-toleading logarithms to all orders in oS using the modified Log(R) matching scheme [S]. A correction for the b-quark mass is calculated at second order using the matrix elements of [9]. The perturbative QCD prediction is corrected for hadronisation and resonance decays by means of a transition matrix, which is computed with Monte Carlo generators. Corrected measurements of event-shape distributions are compared to the theoretical calculations at particle level. The value of CX, is determined at each energy from a binned least-squares fit. Only statistical uncertainties are included in the x2 of the fit. At LEPII it is important to include the ezpetted statistical error rather than the measured error. This eliminates the bias from downward statistical fluctuations in the distributions, which would lead to systematically smaller values of cr,, in particular in the case of ensembles with few events. The same expected error is also assumed for systematic variations of distributions, in order to minimise statistical components in real systematic effects. The results for a, are shown in Figure
290
H. StenzellNuclear ALEPHprdimbq 161
Physics B (Proc. Suppl.) 121 (2003) 59-64
-I
61
ples are available. Uncertainties stem from missing higher orders in the truncated perturbative expansion. A new method is applied to estimate systematic uncertainties related to the perturbative predictions (the ~certainty band method). Sources of these uncertainties are the choice of the renormalization scale (z~) and the logarithmic resealing factor (XL), the matching scheme and the matching modification procedure. A new test is included here for the logarithmic resealing factor XL, which was first proposed for jet broadening in deep-inelastic scattering [lo]. It consist of a replacement of resummed large logarithms L = log(i) (y being an event shape variable such as 1 - T), by a resealed variable L + L = log(&). A variation of 2~ (around unity) is expected to probe missing higher orders in a different way from renormalisation scale variations. Formulae relative to the changes entailed by variations of ZL are given in [8]. The perturbative uncertainties are assessed as follows:
1Q-3 Figure 2. Summary of all individual measurements of (Y, using six variables at eight centre-ofmass energies. The error bars correspond to the total uncertainties, the shaded areas to combined measurements.
l
l
2 for all variables and energies. 3.1.
Experimental
uncertainties
Experimental systematic uncertainties of CY~ are estimated by comparing nominal fit results to fits to distributions obtained under variations of cuts and correction procedures. A special treatment was applied at LEPII for the dominant systematic uncertainty, related to ISR corrections. This uncertainty is subject to fluctuations from one energy to another. An energyindependent luminosity-weighted average is constructed for each variable separately at energies between 133 and 206 GeV, which gives a better estimate of the true systematic effect. All other experimental systematic uncertainties are added in quadrature, yielding a total systematic uncertainty of 0.5-1.5 %. 3.2. Theoretical
Uncertainties
The theoretical uncertainties dominate the total uncertainty at the energies where large sam-
l
l
l
the renormalization tween 0.5 and 2.0,
scale zp is varied be-
the logarithmic resealing factor ZL varied in between 2 /3 and 312 (for - log(ys) an equivalent effect is obtained with squared endpoints, i.e. a variation from 4/9 to 9/4), the modified Log(R) matching scheme is replaced by the modified R matching scheme, the value of the kinematic constraint ylmaz (defined in [S]), obtained with parton shower simulations, is replaced by the value of Yinaz using matrix element calculations and the first degree modification of the modified Log(R) matching scheme (p = 1) is replaced by a second degree modification (p = 2).
The uncertainties in (II, corresponding to these theoretical uncertainties were previously obtained by modifying the theory (e.g., setting a different value of +), and repeating the fits to the data. The uncertainty band method derives directly the uncertainty of (Y, from the uncertainty of
62
H. Stenzel/NucIear
Physics B (Proc. Suppl.) 121 (2003) 59-44
the theoretical prediction. It proceeds in three steps. First a reference perturbative prediction, the modified Log(R) matching scheme, is calculated for the distribution of each variable using the values of crs measured at Mz. Then all variants of the theory mentioned above are calculated with the same value of a,. In each bin of the distribution, the largest upward and downward differences with respect to the reference theory are taken to define an uncertainty band around the reference theory. In the last step, the reference theory is used again, but with a variable CX#,in order to find the range of os values which result into predictions inside the uncertainty band for the fit range under consideration. The largest respectively smallest allowed values of (Y, fulfilling the condition finally set the perturbative systematic error. The method is illustrated in Figure 3 for the case of the heavy jet mass. The theoretical error at LEPII is calculated for each variable separately using the corresponding measurement at LEPI, evolved to the appropriate energy. An additional uncertainty is added in quadrature for the b-quark mass correction procedure. The hadronisation model uncertainty is estimated by using HERWIG [ll] and ARIADNE [12] instead of PYTHIA [13] for both hadronisation and detector corrections. The maximum change with respect to the nominal result using PYTHIA is taken as error. At LEPII energies the hadronisation model uncertainty is again subject to statistical fluctuations. Since non-perturbative effects are expected to decrease with l/Q, the energy evolution of hadronisation errors has been fit to a simple A + B/Q form. The fit was performed for each variable separately. In the fit procedure a weight scaling with luminosity is assigned to the hadronisation uncertainty at each energy point. 4. Results
Complete results for all variables and energies can be found in ref. [8], only combined results can be given here. Measurements from different variables are combined into a single measurement per energy using weighted averages. The weights as-
0.3 1
0.25
sP
0.2
g 9 45
0.15 ,&01
% _a .. 0 P 85 -0.05 I
-0.1 E -0.15 g
km.eotlllrnbl ““caidnly
-0.2
l
,
typka1 fit range
-0.25 -0.3
0
LEF'QCDWGpreliminary 1111,111,111111,1,,,1,,1,,,1, 0.05 0.1 0.15 0.2 0.25
', 0.3
0.35
0.4 Mn
Figure 3. Theoretical uncertainties for the heavy jet mass at LEPI. The shaded area corresponds to the total perturbative uncertainty, the full lines show the extremal predictions obtained with the nominal value of cr8 plus resp. minus its theoretical error. Dotted and dashed lines show various contributions to the theoretical uncertainty.
signed to each variable are chosen proportional to the inverse square of their total errors. All systematic errors are obtained by recomputing the weighted average for all variants of the analysis. Statistical correlations between variables are calculated with Monte Carlo data samples. The combined results on a,(Q) are given in Table 2. A comparison of the measurements with a fit of the 3-100~ QCD evolution is shown in Figure 4. Only the uncorrelated component of the error is included in the fit, which excludes the theoretical uncertainty. The curve is seen to be in excellent agreement with measurements. In a second step combined measurements between 133 and 206 GeV are evaluated at the scale of the Z boson mass. Once all measurements are evolved to the same scale, they can again be combined into a weighted average of a,(&&), again with weights proportional to the inverse square
H. Stenzel/Nuclear
63
Physics B (Pmt. Suppl.) 121 (2003) 59-64
Table 2 Combined results of o*(Q) as obtained with weighted averages. 91.2 0.1201 0.0001 0.0050 0.0016 0.0008 0.0053
stat. error pert. error hadr. error exp.syst. error total error 0.15
IS,
8,
‘,,,,,I
133 0.1175 0.0027 0.0045 0.0011 0.0010 0.0054 I,,
ALEPH
I,,
161 0.1145 0.0042 0.0043 0.0009 0.0010 0.0062
I I,,
17
preliminary
172 0.1091 0.0051 0.0042 0.0008 0.0019 0.0067
183 0.1083 0.0024 0.0039 0.0008 0.0010 0.0047
160
180
200 E,
220 [GeVl
Figure 4. The strong coupling constant o8 measured between 91.2 and 206 GeV. A fit of the 3loop running formula is shown, where the hatched area corresponds to the statistical uncertainty of Am = 245 f 15 MeV.
of total errors. The result is given in Table 3. A combination of measurements at LEPII energies without the point at Mz is given as well. The total uncertainty of the combined LEPII measurement is comparable to the one including LEPI, since the perturbative uncertainties are reduced at higher energies, even after evolution to Mz. The measurements at LEPI and LEPII are in good agreement with each other and with pre-
200 0.1083 0.0015 0.0037 0.0007 0.0010 0.0041
206 0.1048 0.0016 0.0036 0.0007 0.0010 0.0041
Table 3 Weighted average of combined measurements of a,(Mz) obtained at energies from 91.2 GeV to 206 GeV and a combined measurement without the point at Mz. data set as (Mz) stat. error pert. error hadr . error exp. syst. error total error
140
189 0.1102 0.0015 0.0037 0.0007 0.0010 0.0042
viously published [15,16,14,17].
LEPI + II 0.1211 f 0.0009 f 0.0044 f 0.0010 f 0.0011 f 0.0047
ALEPH
LEPII 0.1213 f 0.0011 f 0.0043 f 0.0009 f 0.0013 f 0.0047
measurements
J
of o8
5. Conclusion
New results are presented for a consistent measurement of the strong coupling constant from event-shape variables recorded by ALEPH at centre-of-mass energies between 91.2 GeV and 209 GeV. The energy evolution of the strong coupling constant o, has been investigated and is found to be in good agreement with the QCD expectation. An innovative method has been applied for the first time to assess theoretical uncertainties of perturbative nature. It includes additional tests and derives the uncertainty of os from the uncertainty of the distribution of variables. The new combined ALEPH result is as(Mz) = 0.1211 f 0.0047.
64
H. Stenzel/Nuclenr
Physics B (Proc. SuppI.) 121 (2003) 59-64
REFERENCES 1. D. Decamp et al., ALEPH Collab., Nucl. Instr. Meth. A 294 (1990) 121; D. Decamp et al., ALEPH Collab., Nucl. Instr. Meth. A 360 (1995) 481. 2. ALEPH Collab., ALEPH 1999-023 CONF 1999-018 (1999). 3. S. Brandt et al., Phys. Lett. 12 (1964) 57; E. Farhi, Phys. Rev. Lett. 39 (1977) 1587. 4. T. Chandramohan and L. Clavelli, Nucl. Phys. B184 (1981) 365; L. Clavelli and D. Wyler, Phys. Lett. B103 (1981) 383. 5. P.E.L. Rakow and B.R. Webber, Nucl. Phys. B191 (1981) 63. 6. G. Parisi, Phys. Lett. B74 (1978) 65; J.F. Donoghue, F.E. Low, and S.Y. Pi, Phys. Rev. D20 (1979) 2759. 7. S. Catani et al., Phys. Lett. B269 (1991) 432; W.J. Stirling et al., Proceedings of the Durham Workshop, J. Phys. G: Nucl. Part. Phys. 17 (1991) 1567; N. Brown and W.J. Stirling, Phys. Lett. B252 (1990) 657; S. Bethke et al., Nucl. Phys. B370 (1992) 310. 8. ALEPH Collab., ALEPH 2002-012 CONF 2002-002 (2002). 9. P. Nason and C. Oleari, Nucl. Phys. B521 (1998) 237. 10. M. Dasgupta and G. P. Saiam, DESY-01-160, LPTHE-01-43, hep-ph/0110213 11. G. Marchesini, B.R. Webber, G. Abbiendi, I.G. Knowles, M.H. Seymour, and L. Stance, Comp. Phys. Comm. 67 (1992) 465. 12. L. Liinnblad, Computer Physics Commun. 71 (1992) 15. 13. T. Sjiistrand, Comp. Phys. Comm. 82 (1994) 74. 14. D. Decamp et al., ALEPH Collab., Phys. Lett. B284 (1992) 163. 15. R. Barate et al., ALEPH Collab., Phys. Rep. 294 (1998) 1. 16. D. Buskulic et al., ALEPH Collab., 2. Phys. c73 (1997) 409. 17. R. Barate et al., ALEPH Collab., Z. Phys. C76 (1997) 1.
www.clscvier.com/locatclnpc
Measurements of cx, at Centre-of-Mass Energies between 91 and 209 GeV H. StenzeP * aII. Physikalisches Institut, Universitat GieDen, Heinrich-Buff-Ring 16, D-35392 Giei3en, Germany Consistent measurements of event-shape variables and a, using the complete set of data taken with the ALEPH detector at LEP are presented. The strong coupling constant is determined from a fit of 0(a~)+X’LLA QCD calculations to distributions of six event-shape variables at eight centre-of-mass energies. A new method is proposed to estimate theoretical systematic uncertainties. The results are compared to the expected evolution of
1. Introduction
The most important parameter of perturbative QCD is the coupling constant a,. It can be determined in e+e- annihilations at LEP in a wide range of energies, hence testing its expected energy evolution. The ALEPH experiment has collected data at centre-of-mass energies from 91.2 to 209 GeV. The method to extract Q, has to cope with rather different sample sizes at different energies, spanning from a few hundred events at 161 and 172 GeV to a million events at the Z” boson resonance. Under these conditions, distributions of event-shape variables are the most powerful observables. Event-shape distributions have been calculated to second order in os, these calculations are matched to resummed calculations of leading and next-to-leading logarithms to all orders in as. While this technique has been used before and elsewhere, here a new method is presented to determine theoretical uncertainties of the measurement of crs. A thorough investigation of dominant uncertainties related to missing higher orders has been elaborated. 2. Experimental
Data
Interactions above Mz were recorded at various centre-of-mass energies. The data were accumulated at seven nominal energies given in Table 1, some runs in narrow energy ranges were evolved to the closest nominal energy and lumped to*on
behalf
of the
ALEPH
gether. In addition, a sub-sample of high-quality data at LEPI (Q = E,, = 91.2 GeV) containing 1.1 million events was re-analysed. A detailed description of the ALEPH detector is given in [l]. The @(y) events at LEPII (Q 2 133 GeV) are extracted by selecting hadronic events as described in Ref. [2] in a procedure that has several steps. First hadronic events in which a Z is accompanied by an initial state photon radiation (ISR) are removed. Then various cuts on the event topology and jet configuration are applied to reject background from four-fermion processes (WW, ZZ, Zy’). The resulting distributions of event shapes are finally corrected for residual ISR effects. Irreducible background (5-15 %) is subtracted bin-by-bin. Corrections of detector effects and acceptance are applied yielding distributions at hadron level. A definition of the event-shape variables studied here is given in [2]; these are thrust T [3], heavy jet mass squared M&/s [4], wide and total jet broadening Bw and B-J- [5], C-parameter C [6] and - In y3 [7]. A s an example, the resulting distributions of C-parameter are shown in Figure 1. Experimental systematic uncertainties are obtained with variations of cuts, both for event and track selection, and for background rejection and ISR suppression. The dominant experimental uncertainty at LEPII is related to the modeling of corrections for residual ISR, at LEPI it is the simulation of neutral hadronic calorimeter clusters.
collaboration.
0920-5632/03/S - see front matter doi: 10.1016/SO920-5632(03)01813-9
0 2003 Elsevier
Science
B.V.
All rights
reserved
60
II. Stenzel/Nuclear
Physics
B (Proc,
Suppl.)
121 (2003)
59-64
Table 1 Integrated luminosities and numbers of accepted and expected events. There is an uncertainty the predicted numbers of events. JLdt W-l> 12.30 11.08 9.54 58.83 174.36 206.02 216.19
(2) 133 161 172 183 189 200 206
0
0.1
IIIIIIIIIIIIII.I/,IIIIIIIIIIIII 0.2 0.3 0.4 0.5
events found 806 319 257 1319 3578 3514 3578
0.6
0.7
events expected 822 333 242 1262 3578 3528 3590
:. ,-- , , 0.8 0.9 c 1
Figure 1. The measured distributions, after correction for backgrounds and detector effects, of Cparameter at energies between 91.2 and 206 GeV together with the fitted QCD predictions.
3. Measurements
of cxS
The coupling constant oS is determined from a fit of the perturbative QCD prediction to the Measuremeasured event-shape distributions. ments are performed for each variable and each energy first, then different variables are combined into measurements of a,(Q) and finally all data are combined into a single value of a,(Mz).
expected signal 822 319 218 1109 3124 3005 3072
of 2% in
expected background 0.0 14 24 153 454 523 518
Event-shape distributions are fitted in the central region of 3-jet production, where a good perturbative description is available. The fit range is placed inside a region where hadronisation corrections are well under control, where detector corrections are stable and background corrections are small. The perturbative prediction consists of second order calculations 0(az) which are matched to calculations resumming leading and next-toleading logarithms to all orders in oS using the modified Log(R) matching scheme [S]. A correction for the b-quark mass is calculated at second order using the matrix elements of [9]. The perturbative QCD prediction is corrected for hadronisation and resonance decays by means of a transition matrix, which is computed with Monte Carlo generators. Corrected measurements of event-shape distributions are compared to the theoretical calculations at particle level. The value of CX, is determined at each energy from a binned least-squares fit. Only statistical uncertainties are included in the x2 of the fit. At LEPII it is important to include the ezpetted statistical error rather than the measured error. This eliminates the bias from downward statistical fluctuations in the distributions, which would lead to systematically smaller values of cr,, in particular in the case of ensembles with few events. The same expected error is also assumed for systematic variations of distributions, in order to minimise statistical components in real systematic effects. The results for a, are shown in Figure
290
H. StenzellNuclear ALEPHprdimbq 161
Physics B (Proc. Suppl.) 121 (2003) 59-64
-I
61
ples are available. Uncertainties stem from missing higher orders in the truncated perturbative expansion. A new method is applied to estimate systematic uncertainties related to the perturbative predictions (the ~certainty band method). Sources of these uncertainties are the choice of the renormalization scale (z~) and the logarithmic resealing factor (XL), the matching scheme and the matching modification procedure. A new test is included here for the logarithmic resealing factor XL, which was first proposed for jet broadening in deep-inelastic scattering [lo]. It consist of a replacement of resummed large logarithms L = log(i) (y being an event shape variable such as 1 - T), by a resealed variable L + L = log(&). A variation of 2~ (around unity) is expected to probe missing higher orders in a different way from renormalisation scale variations. Formulae relative to the changes entailed by variations of ZL are given in [8]. The perturbative uncertainties are assessed as follows:
1Q-3 Figure 2. Summary of all individual measurements of (Y, using six variables at eight centre-ofmass energies. The error bars correspond to the total uncertainties, the shaded areas to combined measurements.
l
l
2 for all variables and energies. 3.1.
Experimental
uncertainties
Experimental systematic uncertainties of CY~ are estimated by comparing nominal fit results to fits to distributions obtained under variations of cuts and correction procedures. A special treatment was applied at LEPII for the dominant systematic uncertainty, related to ISR corrections. This uncertainty is subject to fluctuations from one energy to another. An energyindependent luminosity-weighted average is constructed for each variable separately at energies between 133 and 206 GeV, which gives a better estimate of the true systematic effect. All other experimental systematic uncertainties are added in quadrature, yielding a total systematic uncertainty of 0.5-1.5 %. 3.2. Theoretical
Uncertainties
The theoretical uncertainties dominate the total uncertainty at the energies where large sam-
l
l
l
the renormalization tween 0.5 and 2.0,
scale zp is varied be-
the logarithmic resealing factor ZL varied in between 2 /3 and 312 (for - log(ys) an equivalent effect is obtained with squared endpoints, i.e. a variation from 4/9 to 9/4), the modified Log(R) matching scheme is replaced by the modified R matching scheme, the value of the kinematic constraint ylmaz (defined in [S]), obtained with parton shower simulations, is replaced by the value of Yinaz using matrix element calculations and the first degree modification of the modified Log(R) matching scheme (p = 1) is replaced by a second degree modification (p = 2).
The uncertainties in (II, corresponding to these theoretical uncertainties were previously obtained by modifying the theory (e.g., setting a different value of +), and repeating the fits to the data. The uncertainty band method derives directly the uncertainty of (Y, from the uncertainty of
62
H. Stenzel/NucIear
Physics B (Proc. Suppl.) 121 (2003) 59-44
the theoretical prediction. It proceeds in three steps. First a reference perturbative prediction, the modified Log(R) matching scheme, is calculated for the distribution of each variable using the values of crs measured at Mz. Then all variants of the theory mentioned above are calculated with the same value of a,. In each bin of the distribution, the largest upward and downward differences with respect to the reference theory are taken to define an uncertainty band around the reference theory. In the last step, the reference theory is used again, but with a variable CX#,in order to find the range of os values which result into predictions inside the uncertainty band for the fit range under consideration. The largest respectively smallest allowed values of (Y, fulfilling the condition finally set the perturbative systematic error. The method is illustrated in Figure 3 for the case of the heavy jet mass. The theoretical error at LEPII is calculated for each variable separately using the corresponding measurement at LEPI, evolved to the appropriate energy. An additional uncertainty is added in quadrature for the b-quark mass correction procedure. The hadronisation model uncertainty is estimated by using HERWIG [ll] and ARIADNE [12] instead of PYTHIA [13] for both hadronisation and detector corrections. The maximum change with respect to the nominal result using PYTHIA is taken as error. At LEPII energies the hadronisation model uncertainty is again subject to statistical fluctuations. Since non-perturbative effects are expected to decrease with l/Q, the energy evolution of hadronisation errors has been fit to a simple A + B/Q form. The fit was performed for each variable separately. In the fit procedure a weight scaling with luminosity is assigned to the hadronisation uncertainty at each energy point. 4. Results
Complete results for all variables and energies can be found in ref. [8], only combined results can be given here. Measurements from different variables are combined into a single measurement per energy using weighted averages. The weights as-
0.3 1
0.25
sP
0.2
g 9 45
0.15 ,&01
% _a .. 0 P 85 -0.05 I
-0.1 E -0.15 g
km.eotlllrnbl ““caidnly
-0.2
l
,
typka1 fit range
-0.25 -0.3
0
LEF'QCDWGpreliminary 1111,111,111111,1,,,1,,1,,,1, 0.05 0.1 0.15 0.2 0.25
', 0.3
0.35
0.4 Mn
Figure 3. Theoretical uncertainties for the heavy jet mass at LEPI. The shaded area corresponds to the total perturbative uncertainty, the full lines show the extremal predictions obtained with the nominal value of cr8 plus resp. minus its theoretical error. Dotted and dashed lines show various contributions to the theoretical uncertainty.
signed to each variable are chosen proportional to the inverse square of their total errors. All systematic errors are obtained by recomputing the weighted average for all variants of the analysis. Statistical correlations between variables are calculated with Monte Carlo data samples. The combined results on a,(Q) are given in Table 2. A comparison of the measurements with a fit of the 3-100~ QCD evolution is shown in Figure 4. Only the uncorrelated component of the error is included in the fit, which excludes the theoretical uncertainty. The curve is seen to be in excellent agreement with measurements. In a second step combined measurements between 133 and 206 GeV are evaluated at the scale of the Z boson mass. Once all measurements are evolved to the same scale, they can again be combined into a weighted average of a,(&&), again with weights proportional to the inverse square
H. Stenzel/Nuclear
63
Physics B (Pmt. Suppl.) 121 (2003) 59-64
Table 2 Combined results of o*(Q) as obtained with weighted averages. 91.2 0.1201 0.0001 0.0050 0.0016 0.0008 0.0053
stat. error pert. error hadr. error exp.syst. error total error 0.15
IS,
8,
‘,,,,,I
133 0.1175 0.0027 0.0045 0.0011 0.0010 0.0054 I,,
ALEPH
I,,
161 0.1145 0.0042 0.0043 0.0009 0.0010 0.0062
I I,,
17
preliminary
172 0.1091 0.0051 0.0042 0.0008 0.0019 0.0067
183 0.1083 0.0024 0.0039 0.0008 0.0010 0.0047
160
180
200 E,
220 [GeVl
Figure 4. The strong coupling constant o8 measured between 91.2 and 206 GeV. A fit of the 3loop running formula is shown, where the hatched area corresponds to the statistical uncertainty of Am = 245 f 15 MeV.
of total errors. The result is given in Table 3. A combination of measurements at LEPII energies without the point at Mz is given as well. The total uncertainty of the combined LEPII measurement is comparable to the one including LEPI, since the perturbative uncertainties are reduced at higher energies, even after evolution to Mz. The measurements at LEPI and LEPII are in good agreement with each other and with pre-
200 0.1083 0.0015 0.0037 0.0007 0.0010 0.0041
206 0.1048 0.0016 0.0036 0.0007 0.0010 0.0041
Table 3 Weighted average of combined measurements of a,(Mz) obtained at energies from 91.2 GeV to 206 GeV and a combined measurement without the point at Mz. data set as (Mz) stat. error pert. error hadr . error exp. syst. error total error
140
189 0.1102 0.0015 0.0037 0.0007 0.0010 0.0042
viously published [15,16,14,17].
LEPI + II 0.1211 f 0.0009 f 0.0044 f 0.0010 f 0.0011 f 0.0047
ALEPH
LEPII 0.1213 f 0.0011 f 0.0043 f 0.0009 f 0.0013 f 0.0047
measurements
J
of o8
5. Conclusion
New results are presented for a consistent measurement of the strong coupling constant from event-shape variables recorded by ALEPH at centre-of-mass energies between 91.2 GeV and 209 GeV. The energy evolution of the strong coupling constant o, has been investigated and is found to be in good agreement with the QCD expectation. An innovative method has been applied for the first time to assess theoretical uncertainties of perturbative nature. It includes additional tests and derives the uncertainty of os from the uncertainty of the distribution of variables. The new combined ALEPH result is as(Mz) = 0.1211 f 0.0047.
64
H. Stenzel/Nuclenr
Physics B (Proc. SuppI.) 121 (2003) 59-64
REFERENCES 1. D. Decamp et al., ALEPH Collab., Nucl. Instr. Meth. A 294 (1990) 121; D. Decamp et al., ALEPH Collab., Nucl. Instr. Meth. A 360 (1995) 481. 2. ALEPH Collab., ALEPH 1999-023 CONF 1999-018 (1999). 3. S. Brandt et al., Phys. Lett. 12 (1964) 57; E. Farhi, Phys. Rev. Lett. 39 (1977) 1587. 4. T. Chandramohan and L. Clavelli, Nucl. Phys. B184 (1981) 365; L. Clavelli and D. Wyler, Phys. Lett. B103 (1981) 383. 5. P.E.L. Rakow and B.R. Webber, Nucl. Phys. B191 (1981) 63. 6. G. Parisi, Phys. Lett. B74 (1978) 65; J.F. Donoghue, F.E. Low, and S.Y. Pi, Phys. Rev. D20 (1979) 2759. 7. S. Catani et al., Phys. Lett. B269 (1991) 432; W.J. Stirling et al., Proceedings of the Durham Workshop, J. Phys. G: Nucl. Part. Phys. 17 (1991) 1567; N. Brown and W.J. Stirling, Phys. Lett. B252 (1990) 657; S. Bethke et al., Nucl. Phys. B370 (1992) 310. 8. ALEPH Collab., ALEPH 2002-012 CONF 2002-002 (2002). 9. P. Nason and C. Oleari, Nucl. Phys. B521 (1998) 237. 10. M. Dasgupta and G. P. Saiam, DESY-01-160, LPTHE-01-43, hep-ph/0110213 11. G. Marchesini, B.R. Webber, G. Abbiendi, I.G. Knowles, M.H. Seymour, and L. Stance, Comp. Phys. Comm. 67 (1992) 465. 12. L. Liinnblad, Computer Physics Commun. 71 (1992) 15. 13. T. Sjiistrand, Comp. Phys. Comm. 82 (1994) 74. 14. D. Decamp et al., ALEPH Collab., Phys. Lett. B284 (1992) 163. 15. R. Barate et al., ALEPH Collab., Phys. Rep. 294 (1998) 1. 16. D. Buskulic et al., ALEPH Collab., 2. Phys. c73 (1997) 409. 17. R. Barate et al., ALEPH Collab., Z. Phys. C76 (1997) 1.