1. Joint Institute for Nuclear Research, P O Box 79, 141980 Dubna, Russia;
2. University of Oklahoma, Department of Physics and Astronomy, Norman, Oklahoma 73019, USA;
3. Laboratoire de l'Accélérateur Linéaire, IN2P3-CNRS, BP 34, Batiment 200, F-91898 Orsay, France;
4. Michigan State University, Department of Physics and Astronomy, East Lansing, Michigan 48824, USA;
5. Institute for High Energy Physics, 142284 Protvino, Russia;
6. Tata Institute of Fundamental Research, School of Natural Sciences, Homi Bhabha Rd, Mumbai 400005, India;
7. Brookhaven National Laboratory, Physics Department, Bldg 510C, Upton, New York 11973, USA;
8. University of Illinois at Chicago, Department of Physics, 845 W. Taylor, Chicago, Illinois 60607, USA;
9. University of Nijmegen/NIKHEF, P O Box 9010, NL-6500 GL Nijmegen, The Netherlands;
10. University of Michigan, Department of Physics, 500 E. University Avenue, Ann Arbor, Michigan 48109, USA;
11. LAFEX, Centro Brasileiro de Pesquisas Físicas, Rua Dr Xavier Sigaud, 150, 22290-180 Rio de Janeiro, Brazil;
12. Laboratoire de Physique Subatomique et de Cosmologie, IN2P3-CNRS, Université de Grenoble 1, 53 Avenue des Martyrs, F-38026 Grenoble, France;
13. Universidad de los Andes, Department de Fisica, HEP Group, Apartado Aereo 4976, Bogotá, Colombia;
14. State University of New York, Department of Physics and Astronomy, Stony Brook, New York 11794, USA;
15. Imperial College London, Department of Physics, Prince Consort Road, London SW7 2BW, UK;
16. University of Maryland, Department of Physics, College Park, Maryland 20742, USA;
17. CPPM, IN2P3-CNRS, Université de la Méditerranée, 163 Avenue de Luminy, F-13288 Marseille, France;
18. Fermi National Accelerator Laboratory, P O Box 500, Batavia, Illinois 60510, USA;
19. FOM-Institute NIKHEF and University of Amsterdam/NIKHEF, P O Box 41882, 1009 DB Amsterdam, The Netherlands;
20. Northeastern University, Department of Physics, Boston, Massachusetts 02115, USA;
21. University of Kansas, Department of Physics and Astronomy, 1251 Wescoe Hall Drive, Lawrence, Kansas 66045, USA;
22. LPNHE, Universités Paris VI and VII, IN2P3-CNRS, 4 Place Jussieu, Tour 33, F-75252 Paris, France;
23. Indiana University, Department of Physics, 727 E. 3rd St, Bloomington, Indiana 47405, USA;
24. University of Rochester, Department of Physics and Astronomy, Rochester, New York 14627, USA;
25. Florida State University, Department of Physics 4350, Tallahassee, Florida 32306, USA;
26. Panjab University, Department of Physics, Chandigarh 160014, India;
27. Lancaster University, Department of Physics, Lancaster LA1 4YB, United Kingdom;
28. Northern Illinois University, Department of Physics, DeKalb, Illinois 60115, USA;
29. Kansas State University, Department of Physics, Manhattan, Kansas 66506, USA;
30. Columbia University, Department of Physics, 538 W. 120th St, New York, New York 10027, USA;
31. University of Texas, Department of Physics, Box 19059, Arlington, Texas 76019, USA;
32. Brown University, Department of Physics, 182 Hope St, Providence, Rhode Island 02912, USA;
33. Northwestern University, Department of Physics and Astronomy, 2145 Sheridan Road, Evanston, Illinois 60208, USA;
34. Universität Freiburg, Physikalisches Institut, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany;
35. Boston University, Department of Physics, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA;
36. Universidade do Estado do Rio de Janeiro, Instituto de Física, Rua São Francisco Xavier, 524, 20559-900 Rio de Janeiro, Brazil;
37. CINVESTAV, Departamento de Física, P O Box 14-740, 07000 Mexico City, Mexico;
38. University of California, Department of Physics, Riverside, California 92521, USA;
39. University of Nebraska, Department of Physics and Astronomy, Lincoln, Nebraska 68588, USA;
40. Lawrence Berkeley National Laboratory and University of California, 1 Cyclotron Road, Berkeley, California 94720, USA;
41. DAPNIA/Service de Physique des Particules, CEA, Saclay, F-91191 Gif-sur-Yvette, France;
42. Moscow State University, Department of Physics, Vorobjovy Gory, 119899 Moscow, Russia;
43. Rice University, Bonner Nuclear Lab, P O Box 1892, Houston, Texas 77005, USA;;
44. Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya ul. 25, 117259 Moscow, Russia;
45. University of Notre Dame, Department of Physics, Notre Dame, Indiana 46556, USA;
46. Louisiana Tech University, Department of Physics, Ruston, Louisiana 71272, USA;;
47. Universidad de Buenos Aires, Departamento de Física, FCEN, Pabellón 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina;
48. University College Dublin, Department of Experimental Physics, Faculty of Science, Belfield, Dublin 4, Ireland;
49. California State University, Department of Physics, 2345 E. San Ramon Avenue, Fresno, California 93740, USA;
50. Iowa State University, Department of Physics, High Energy Physics Group, Ames, Iowa 50011, USA;
51. University of Virginia, Department of Physics, Charlottesville, Virginia 22901, USA;
52. Universidad San Francisco de Quito, P O Box 17-12-841, Quito, Ecuador;
53. University of Arizona, Department of Physics, P O Box 210081, Tucson, Arizona 85721, USA;
54. Institute of Particle and Nuclear Physics, Center for Particle Physics, Faculty of Mathematics and Physics, Charles University in Prague, V Holesovickach 2, CZ-18000 Prague 8, Czech Republic;
55. Institute of Physics of the Academy of Sciences of the Czech Republic, Center for Particle Physics, Na Slovance 2, CZ-18221 Prague 8, Czech Republic;
56. Institute of High Energy Physics, P O Box 918, Beijing 100039, China;
57. Langston University, Department of Mathematics, Langston, Oklahoma 73050, USA;
58. Delhi University, Department of Physics and Astrophysics, Delhi 110007, India;
59. University of California, Department of Physics and Astronomy, 4129 Frederick Reines Hall, Irvine, California 92697, USA;
60. University of Washington, Department of Physics, P O Box 351560, Seattle, Washington 98195, USA.
61. Present addresses: University of Zurich, Zurich, Switzerland (F.L.); Institute of Nuclear Physics, Krakow, Poland (M.B.P.)

The discovery of the top quark in 1995 served as one of the major confirmations of the validity of the standard model (SM)^{7, 8}. Of its many parameters, the mass of the top quark, in particular, reflects some of the most crucial aspects of the SM. This is because, in principle, the top quark is point-like and should be massless; yet, through its interactions with the hypothesized Higgs field, the physical mass of the top quark appears to be about the mass of a gold nucleus. Because it is so heavy, the top quark (along with the W boson) provides an unusually sensitive tool for investigating the Higgs field. M_W is known to a precision of 0.05%, while the uncertainty on M_t is at the 3% level¹. Improvements in both measurements are required to restrict further the allowed range of mass for the Higgs; however, given the large uncertainty in M_t, an improvement in its precision is particularly important. As has been pointed out recently^{9, 10}, a potential problem for the SM is that, on the basis of the currently accepted value for M_t, the most likely value of the Higgs mass⁶ lies in a range that has already been excluded by experiment⁵. Precise knowledge of the Higgs mass is crucial for our understanding of the SM and any possible new physics beyond it. For example, in a large class of supersymmetric models (theoretically preferred solutions to the deficiencies of the SM), the Higgs mass has to be less than about 135 GeV/c². Although, unlike the SM, supersymmetry predicts more than one Higgs boson, the properties of the lightest one are expected to be essentially the same as those for the SM Higgs boson. Thus, if the SM-like Higgs is heavier than about 135 GeV/c², it would disfavour a large class of supersymmetric models. In addition, some of the current limits on supersymmetric particles from LEP¹¹ are extremely sensitive to M_t. In fact, for M_t greater than 179 GeV/c², the bounds on one of the major supersymmetry parameters, tan, which relates the properties of the SM-like Higgs boson and its heavier partners, would disappear completely¹². Hence, in addition to the impact on searches for the Higgs boson, other important consequences call for improved precision on M_t, and this goal is the main subject of this paper.

The DØ experiment at the Fermilab Tevatron has studied a sample of t events produced in proton–antiproton (p) interactions¹³. The total energy of 1.8 TeV released in a head-on collision of a 900-GeV p and a 900-GeV is almost as large as the rest energy of ten gold nuclei. Each top (antitop) quark decays almost immediately into a bottom b() quark and a W⁺ (W ^-) boson, and we have reexamined those events in which one of the W bosons decays into a charged lepton (electron or muon) and a neutrino, and the other W into a quark and an antiquark (see Fig. 1). These events and their selection criteria are identical to those used to extract the mass of the top quark in our previous publication, and correspond to an integrated luminosity of 125 events per pb. (That is, given the production cross-section of the t in p collisions at 1.8 TeV of 5.7 pb, as measured by Dع⁴, these data correspond to approximately 700 produced t pairs, a fraction of which is fully detected in various possible decay modes. Approximately 30% of these correspond to the lepton + jets topology categorized in Fig. 2, where 'jet' refers to products of the fragmentation of a quark into a collimated group of particles that are emitted along the quark's original direction.) The main background processes correspond to multijet production (20%), where one of the jets is reconstructed incorrectly as a lepton, and the W + jets production with leptonic W decays (80%), which has the same topology as the t signal.

Figure 1: Feynman diagrams for t production in p collisions, with subsequent decays into an electron, neutrino, and quarks.

Quark–antiquark production (a) is dominant, but gluon fusion (b) contributes 10% to the cross-section. This particular final state (eub) is one of the channels used in the analysis.

High resolution image and legend (23K)

Figure 2: Relative importance of various t decay modes.

The 'lepton + jets' channel used in this analysis corresponds to the two offset slices of the pie-chart and amounts to 30% of all the t decays.

High resolution image and legend (33K)

The previous DØ measurement of M_t in this lepton + jets channel is M_t = 173.3 5.6 (stat) 5.5 (syst) GeV/c², and is based on 91 candidate events. Information pertaining to the older analysis and the DØ detector can be found elsewhere^{13, 15}.

The new method of M_t measurement is similar to one suggested previously (ref. 16 and references therein, and ref. 17) for t dilepton decay channels (where both W bosons decay leptonically), and used in previous mass analyses of dilepton events³, and akin to an approach suggested for the measurement of the mass of the W boson at LEP^{18, 19, 20}. The critical differences from previous analyses in the lepton + jets decay channel lie in: (1) the assignment of more weight to events that are well measured or more likely to correspond to t signal, and (2) the handling of the combinations of final-state objects (lepton, jets and imbalance in transverse momentum, the latter being a signature for an undetected neutrino) and their identification with top-quark decay products in an event (such as from ambiguity in choosing jets that correspond to b or quarks from the decays of the t and quarks). Also, because leading-order matrix elements were used to calculate the event weights, only events with exactly four jets are kept in this analysis, resulting in a candidate sample of 71 events. Although we are left with fewer events, the new method for extracting M_t provides substantial improvement in both statistical and systematic uncertainties.

We calculate as a function of M_t the differential probability that the measured variables in any event correspond to signal. The maximum of the product of these individual event probabilities provides the best estimate of M_t in the data sample. The impact of biases from imperfections in the detector and event-reconstruction algorithms is taken into account in two ways. Geometric acceptance, trigger efficiencies, event selection, and so on enter through a multiplicative acceptance function that is independent of M_t. Because the angular directions of all the objects in the event, as well as the electron momentum, are measured with high precision, their measured values are used directly in the calculation of the probability that any event corresponds to t or background production. The known momentum resolution is used to account for uncertainties in measurements of jet energies and muon momenta.

As in the previous analysis¹³, momentum conservation in + jet events is used to check that the energies of jets in the experiment agree with Monte Carlo (MC) simulations. This calibration has an uncertainty E = (0.025E + 0.5 GeV). Consequently, all jet energies in our sample are rescaled by E, the analysis redone, and half of the difference in the two rescaled results for M_t (M_t = 3.3 GeV/c²) is taken as a systematic uncertainty from this source. All other contributions to systematic uncertainty: MC modelling of signal (M_t = 1.1 GeV/c²) and background (M_t = 1.0 GeV/c²), effect of calorimeter noise and event pile-up (M_t = 1.3 GeV/c²), and other corrections from M_t extraction (M_t = 0.6 GeV/c²) are much smaller, and discussed in detail elsewhere^{21, 22}. It should be noted that the new mass measurement method provides a significant (about 40%, from 5.5 to 3.9 GeV/c²) reduction in systematic uncertainty, which is ultimately dominated by the measurement of jet energies. For details on the new analysis, see the Methods.

The final result is M_t = 180.1 3.6 (stat) 3.9 (syst) GeV/c². The improvement in statistical uncertainty over our previous measurement is equivalent to collecting a factor of 2.4 as much data. Combining the statistical and systematic uncertainties in quadrature, we obtain M_t = 180.1 5.3 GeV/c², which is consistent with our previous measurement in the same channel (at about 1.4 standard deviations), and has a precision comparable to all previous M_t measurements combined¹.

The new measurement can be combined with that obtained for the dilepton sample that was also collected at DØ during run I (ref. 3) (M_t = 168.4 12.3 (stat) 3.6 (syst) GeV/c²), to yield the new DØ average for the mass of the top quark:

Combining this with measurements from the CDF experiment² provides a new 'world average' (based on all measurements available) for the mass of the top quark⁴:

dominated by our new measurement. This new world average shifts the best-fit value of the expected Higgs mass from 96 GeV/c² to 117 GeV/c² (see Fig. 3), which is now outside the experimentally excluded region, yet accessible in the current run of the Tevatron and at future runs at the Large Hadron Collider (LHC), at present under construction at CERN. (The upper limit on the Higgs mass at the 95% confidence level changes from 219 GeV/c² to 251 GeV/c².) Figure 3 shows the effect of using only the new DØ top mass for fits to the Higgs mass, and indicates a best value of 123 GeV/c² and the upper limit of 277 GeV/c² at the 95% confidence level. It should be noted that the horizontal scale in Fig. 3 is logarithmic, and the limits on the Higgs boson mass are therefore asymmetric.

Figure 3: Current experimental constraints on the mass of the Higgs boson.

The ² for a global fit to electroweak data⁶ is shown as a function of the Higgs mass. The solid line corresponds to the result for the previous world-averaged M_t = 174.3 5.1 GeV/c², with the blue band indicating the impact of theoretical uncertainty. The dotted line shows the result for the new world-averaged M_t of 178.0 4.3 GeV/c², whereas the dashed line corresponds to using only the new DØ average of 179.0 5.1 GeV/c². The yellow-shaded area on the left indicates the region of Higgs masses excluded by experiment (>114.4 GeV/c² at the 95% confidence level⁵). The improved M_t measurement shifts the most likely value of the Higgs mass above the experimentally excluded range.

High resolution image and legend (80K)

The new method is already being applied to data being collected by the CDF and DØ experiments at the new run of the Fermilab Tevatron and should provide even higher precision on the determination of M_t, equivalent to more than a doubling of the data sample, relative to using the conventional method. An ultimate precision of about 2 GeV/c² on the mass of the top quark is expected to be reached in several years of Tevatron operation. Further improvement may eventually come from the LHC.

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Methods

The probability density as a function of M_t can be written as a convolution of the calculable cross-section and any effects from measurement resolution:

where W(y, x), our general transfer function, is the normalized probability for the measured set of variables x to arise from a set of nascent (partonic) variables y, d(y, M_t) is the partonic theoretical differential cross-section, f(q) are parton distribution functions that reflect the probability of finding any specific interacting quark (antiquark) with momentum q within the proton (antiproton), and (M_t) is the total cross-section for producing t. The integral in equation (3) sums over all possible parton states, leading to what is observed in the detector.

The acceptance of the detector is given in terms of a function A(x) that relates the probability P_m(x, M_t) of measuring the observed variables x to their production probability P(x, M_t): P_m(x, M_t) = A(x)P(x, M_t). Effects from energy resolution, and so on are taken into account in the transfer function W(y, x). The integrations in equation (3) over the eleven well-measured variables (three components of charged-lepton momentum and eight jet angles) and the four equations of energy-momentum conservation leave five integrals that must be performed to obtain the probability that any event represents t (or background) production for some specified value of M_t.

The probability for a t interpretation can be written as:

where represents a set of five integration variables, M_t is the leading-order matrix element for t production^{23, 24}, f(q₁) and f(q₂) are the CTEQ4M parton distribution functions for the incident quarks²⁵, ₆ is the phase-space factor for the six-object final state, and the sum is over all 12 permutations of the jets and all possible neutrino solutions. W_jets(E_part, E_jet) corresponds to a function that maps parton-level energies E_part to energies measured in the detector E_jet and is based on MC studies. A similar expression, using a matrix element for W + jets production (the dominant background source) that is independent of M_t, is used to calculate the probability for a background interpretation, P_bkg.

Studies of samples of HERWIG (ref. 26; we used version 5.1) MC events indicate that the new method is capable of providing almost a factor-of-two reduction in the statistical uncertainty on the extracted M_t. These studies also reveal that there is a systematic shift in the extracted M_t that depends on the amount of background there is in the data. To minimize this effect, a selection is introduced, based on the probability that an event represents background. The specific value of the P_bkg cut-off is based on MC studies carried out before applying the method to data, and, for M_t = 175 GeV/c², retains 71% of the signal and 30% of the background. A total of 22 data events out of our 71 candidates pass this selection.

The final likelihood as a function of M_t is written as:

The integration is performed using MC methods. The best value of M_t (when L is at its maximum L_max) represents the most likely mass of the top quark in the final N-event sample, and the parameters c_i reflect the amounts of signal and background. MC studies show that there is a downward shift of 0.5 GeV/c² in the extracted mass, and this correction is applied to the result. Reasonable changes in the cut-off on P_bkg do not have a significant impact on M_t.

Figure 4 shows the value of L(M_t)/L_max as a function of M_t for the 22 events that pass all selection criteria, after correction for the 0.5 GeV/c² bias in mass. The likelihood is maximized with respect to the parameters c_i at each mass point. The gaussian fit in the figure yields M_t = 180.1 GeV/c², with a statistical uncertainty of M_t = 3.6 GeV/c². The systematic uncertainty, dominated by the measurement of jet energies, as discussed above, amounts to M_t = 3.9 GeV/c². When added in quadrature to the statistical uncertainty from the fit, it yields the overall uncertainty on the new M_t measurement of 5.3 GeV/c².

Figure 4: Determination of the mass of the top quark using the maximum-likelihood method.

The points represent the likelihood of the fit used to extract M_t divided by its maximum value, as a function of M_t (after a correction for a -0.5 GeV/c² mass bias, see text). The solid line shows a gaussian fit to the points. The maximum likelihood corresponds to a mass of 180.1 GeV/c², which is the new DØ measurement of M_t in the lepton + jets channel. The shaded band corresponds to the range of 1 standard deviation, and indicates the 3.6 GeV/c² statistical uncertainty of the fit.

High resolution image and legend (19K)

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Acknowledgements

We are grateful to our colleagues A. Quadt and M. Mulders for reading of the manuscript and comments. We thank the staffs at Fermilab and collaborating institutions, and acknowledge support from the Department of Energy and National Science Foundation (USA), Commissariat à L'Energie Atomique and CNRS/Institut National de Physique Nucléaire et de Physique des Particules (France), Ministry for Science and Technology and Ministry for Atomic Energy (Russia), CAPES, CNPq and FAPERJ (Brazil), Departments of Atomic Energy and Science and Education (India), Colciencias (Colombia), CONACyT (Mexico), Ministry of Education and KOSEF (Korea), CONICET and UBACyT (Argentina), The Foundation for Fundamental Research on Matter (The Netherlands), PPARC (UK), Ministry of Education (Czech Republic), the A. P. Sloan Foundation, and the Research Corporation.Authors' contributions We wish to note the great number of contributions made by the late Harry Melanson to the DØ experiment, through his steady and inspirational leadership of the physics, reconstruction and algorithm efforts.

Correspondence and requests for materials should be addressed to J. Estrada (Email: estrada@fnal.gov).

Competing interests statement

The author declares no competing financial interests.

DØ Collaboration

DØ Collaboration (Participants are listed in alphabetical order.)

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