Ratios of Multijet Cross Sections in p-pbar Collisions at sqrt(s)=1.8 TeV

We report on a study of the ratio of inclusive three-jet to inclusive two-jet production cross sections as a function of total transverse energy in p-pbar collisions at a center-of-mass energy sqrt{s} = 1.8 TeV, using data collected with the D0 detector during the 1992-1993 run of the Fermilab Tevatron Collider. The measurements are used to deduce preferred renormalization scales in perturbative O(alpha_s^3) QCD calculations in modeling soft-jet emission.

We report on a study of the ratio of inclusive three-jet to inclusive two-jet production cross sections as a function of total transverse energy in pp collisions at a center-of-mass energy √ s = 1.8 TeV, using data collected with the DØ detector during the 1992-1993 run of the Fermilab Tevatron Collider.The measurements are used to deduce preferred renormalization scales in perturbative O(α 3 s ) QCD calculations in modeling soft-jet emission.
A primary manifestation of Quantum Chromodynamics (QCD) in pp collisions at a high center-of-mass energy ( √ s = 1.8 TeV) is the production of jets with large transverse momenta.Typically, the hard interaction of parton constituents of a proton and an antiproton produce two hard back-to-back jets.However, a fraction of the time, additional jets are also produced.In the absence of an all-orders QCD calculation, jet production rates as a function of jet energy are predicted by fixed-order calculations in perturbative QCD (pQCD).In this paper, we investigate the dependence of these calculations on the choice of parton distribution functions (pdf) and particularly renormalization and factorization scales We examine the ratio of inclusive three-jet production to inclusive two-jet production, which reflects the rate of gluon emission in QCD jet production processes.A three jet cross section explicitly offers the opportunity to investigate a scale difference at a secondary vertex.Taking the ratio reduces systematic uncertainties.
Although this issue has inherent theoretical interest, it is also important because QCD multijet production is frequently a background to rare processes: phenomenologically confirmed prescriptions for renormalization scales are essential for predicting background rates and for designing efficient triggering schemes for rare processes at future colliders [1].Lastly, when higher order QCD calculations become available, this ratio may be useful for providing another accurate measure of the strong coupling constant α s .
The data used in this analysis, corresponding to an integrated luminosity of ≈ 10 pb −1 , were recorded during the 1992-1993 Tevatron collider run.The DØ detector is described in detail elsewhere [2].Jet detection primarily utilizes the uranium-liquid argon calorimeters, which have full coverage for pseudorapidity |η| ≤ 4 where η = −ln[tan(θ/2)] and θ is the polar angle relative to the direction of the proton beam.Initial event selection occurred in two hardware trigger stages and a software stage.The first hardware trigger selected an inelastic pp collision as indicated by signals from trigger hodoscopes located near the beams on either side of the interaction region.The next stage required transverse energy above a preset threshold in calorimeter towers of 0.2 × 0.2 in ∆η ×∆φ, where φ is the azimuthal angle.Selected events were digitized and sent to an array of processors.Jet candidates were then reconstructed with a cone algorithm and the event recorded if any jet transverse energy (E T ) exceeded a specified threshold.Five such inclusive triggers had thresholds of 20, 30, 50, 85, and 115 GeV.
Jets were reconstructed offline using an iterative fixedcone algorithm with a cone radius R = 0.7 in η − φ space.The E T of each jet was corrected for effects due to the underlying event, additional interactions, noise from uranium decay, the fraction of particle energy deposited outside of the reconstruction cone, detector uniformity, and detector hadronic response.A discussion of the jet algorithm, energy scale calibration and resolution can be found in Refs.[3][4][5].
We measure the ratio of the inclusive three-jet to the inclusive two-jet cross section as a function of the scalar sum of jet transverse energies (H T = E jet T ).The measurement is performed for four distinct sets of selection criteria for all jets in the event: E T thresholds of 20, 30, or 40 GeV for |η jet | < 3, and E T > 20 GeV for |η jet | < 2. Three thresholds were chosen to study threshold dependence, and the minimum threshold was chosen to maximize statistics for which jet reconstruction efficiency was nearly 100%.Both in the data analysis and in the QCD calculation, a jet contributes to H T and to the jet multiplicity if it passes all selection criteria and satisfies the E T and η jet requirements.
Figure 1 shows the ratio R 32 as a function of H T for E T thresholds of 20, 30, and 40 GeV for |η jet | < 3. The five trigger samples listed in the figure contribute in separate regions of H T , as indicated by the symbols.The distribution at the bottom of the figure shows the correlated systematic uncertainties for the 20 GeV threshold.This uncertainty is the maximum offset in the ratio obtained by a one standard deviation change in the correction to the jet energy scale.Error bars indicate statistical uncertainties (calculated using the appropriate binomial prescription for a statistically correlated ratio) as well as uncorrelated systematic uncertainties arising from all selection criteria.Table I displays the measurements in bins of H T , showing the uncorrelated and correlated uncertainties for the four selection criteria.
jetrad [6] is a next-to-leading-order Monte Carlo generator for describing inclusive multijet production.The generated 2-jet and 3-jet events are inclusive, and therefore the ratio of these cross sections should be equivalent to the measured R 32 .CTEQ4M [7] pdf are used in the jetrad simulations.The jet finding algorithm in jetrad approximates the algorithm used in DØ data reconstruction.Jets generated by jetrad are individually smeared according to known detector resolutions.Two partons are combined if they are within R sep = 1.3R, as motivated by the separation of jets in the data [8] and, just as in the data, a jet is included if its E T and η jet meet the chosen selection criteria.
In pQCD, the renormalization procedure introduces a mass scale µ R to control ultraviolet divergences in the calculations.A factorization scale µ F , introduced to han-dle infrared divergences, is assumed to be equal to µ R in all predictions described in this paper.QCD provides the evolution of α s with µ R , but not its absolute scale.Unless otherwise indicated, the renormalization scale µ R = λH T will be used for the production of the two leading jets, where the constant λ, the coefficient of the hard scale, will have a nominal value of 0.3, but will be allowed to vary as described below.To study the possibility of having a different scale for the production of additional jets, the renormalization scale of the third jet is varied from µ (3) R = λH T (same as for the leading jets) to a scale proportional to the E T of the third jet µ T .Also, a scale proportional to the maximum jet transverse energy (E max T ) is studied, as this is a standard form used for comparisons of jetrad to measured jet cross sections.
Figure 2 shows the measured R 32 as a function of H T for jet E T > 20 GeV and |η jet | < 2. The 20 GeV threshold has good sensitivity to scale in the jetrad prediction and has reduced statistical uncertainty.The central rapidity region has the best understood jet energy uncertainties and correlations.The plot contains four smoothed distri- butions corresponding to jetrad predictions for the following renormalization prescriptions (shown for λ = 0.3): • µ R = λH T for the two leading jets, -µ R = λH T also for the third jet (solid) T for the third jet (dashed) -µ T for the third jet (dotted) • µ R = 0.6E max T for all jets (dash-dot).
All predictions demonstrate the same qualitative behavior as the R 32 measurement, that is, a rapid rise below H T = 200 GeV (associated with the kinematic threshold), a leveling off, then a slight drop at highest H T (associated with the reduced phase space for additional radiation for high E T jets).Although jetrad predictions for the ratio are found to be insensitive to the choice of pdf, they do depend on the choice of R sep .Allowing R sep to vary such that neighboring jets are all merged or all split causes a 3% decrease or increase in the ratio, respectively, with only a slight effect on the shape of the distribution in H T .
For a quantitative comparison, we use a χ 2 covariance technique, defining where D i and T i represent the i th data and theory element, respectively, and C −1 is the inverse of the covariance matrix.This matrix incorporates uncorrelated uncertainties in the measurement and statistical uncertainties in the simulation, with correlated uncertainties included for the absolute jet energy in the data and for the uncertainty from resolution smearing in jetrad (not shown explicitly in Fig. 2).Although some of the predictions do not visually overlap with the data, acceptable agreement is found for some scales because of the strong point-to-point correlations of the data uncertainties which are taken into account in the χ 2 .Figure 3 shows the χ 2 per degree of freedom (χ 2 /dof) as a function of the parameter λ, for the E T > 20 GeV, |η jet | < 2 selection criteria.The degrees-of-freedom equal the number of data points (28).The horizontal line indicates the χ 2 /dof obtained using the λ independent scale µ R = 0.6E max T for all jets.This scale yields good agreement with measurement (probability p > 57%) for the E T > 20 GeV criteria, but the χ 2 rises (and the corresponding probabilities decrease) for the higher E T thresholds (not shown).
For λ-dependent scales, the best fit is specified by the λ that minimizes the χ 2 .The scales proportional to E (3) T for the third jet do not provide a good fit (p < 5%) for any λ, as seen in Fig. 3.While there is fair agreement in the wider region of pseudorapidity |η jet | < 3 for certain regions of λ (not shown), these do not correspond to the same values for different E T thresholds, making the applicability of this scale prescription unsuitable for predicting production rates for additional jets.
The jetrad prediction assuming a scale µ R = λH T provides the best description of the data for λ between 0.30 and 0.35 (p > 80%).Moreover, the χ 2 is also minimized in the λ ≈ 0.30 region for the other selection criteria (not shown) making this scale choice the most robust of all the µ R scales studied.
In conclusion, we have measured the ratio of the inclusive three-jet to the inclusive two-jet cross section as a function of total scalar transverse energy H T and compared the results to jetrad predictions.The greatest sensitivity to the choice of renormalization scale is for the lowest E T threshold of 20 GeV.Although no prediction accurately describes the ratio through the kinematic threshold region, a single µ R scale assumption in the calculation for all jets is found to adequately describe the rate of additional jet emission when correlated uncertainties are accounted for in a χ 2 comparison.Specifically, a scale of µ R = λH T for all jets, where λ = 0.3, yields a prediction consistent with the measurement for all jetselection criteria examined.A scale of µ R = 0.6E max T for all jets also provides a sufficient description at the lowest jet E T threshold.The introduction of additional scales does not significantly improve agreement with the data.

FIG. 1 .
FIG. 1.The ratio R32 as a function of HT for ET thresholds of 20, 30, and 40 GeV (|ηjet| < 3).Error bars indicate statistical and uncorrelated systematic uncertainties, while the distribution at the bottom shows the correlated systematic uncertainty for the 20 GeV threshold.

FIG. 2 .
FIG. 2.The ratio R32 as a function of HT , requiring jet ET > 20 GeV and |ηjet| < 2. Error bars indicate statistical and uncorrelated systematic uncertainties, while the histogram at the bottom shows the correlated systematic uncertainty.The four smoothed distributions show the jetrad prediction for the renormalization scales indicated in the legend.

TABLE I .
Values of R32 with their uncorrelated and correlated uncertainties for the indicated jet ET threshold and ηjet criteria.Uncorrelated uncertainties include statistical and uncorrelated systematic uncertainties added in quadrature.