(T/E) Variation of the Strong Coupling Constant from a Measurement of the Jet Energy Spread in e+e ' Annihilation *

Abstract A measurement of jet energy spread in the reaction e+e− → hadrons is presented. Using a jet calculus model for the jet development we determine the variation of the strong coupling constant with respect to momentum transfer. The observed variation is consistent with that expected for QCD over a wide range of momentum transfers. This method alone is not sufficient to distinguish QCD from simple limited transverse momentum models.

In recent years the theory of the strong interaction, QCD, has been successful in explaining the characteristics of deep inelastic lepton-nucleon scattering and hadron production in e+e -annihilation. An important consequence of QCD is a decreasing coupling constant with increasing energy. The experimental verification of this fact is difficult, since the coupling constant a s changes logarithmically with the energy. The value ofa s at fluxed energy can be determined by the leptonic branching ratios of the qJ, q/and T resonances [ 1 ]. Measurements of a s over a range of energies in deep inelastic lepton-nucleon scattering have large statistical errors [2]. Konishi et al. [3] have suggested a statistically powerful method which uses the angular energy spread inside a hadron jet to determine a s . In jet development, the relevant mass scale in the successive branching of partons varies from half the center-of-mass energy down to a few GeV, thus allowing the variation of the effective coupling constant to be determined over almost two orders of magnitude inq2.
In this method, energy and momenta are measured using a set of fictitious calorimeters that completely cover a jet produced in the reaction e+e -~ hadrons. Each calorimeter subtends an opening angle (26). If E i is the energy measured in the ith calorimeter, than the jet energy is given by N = ~ Ei (6 ) , N = number of calorimeters ej et and the jet energy spread of order n is defined as 90 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland cn(g)=(~xn (8)) withxi =Ei(f)/~Ej (5 ) . (1) The mean value is computed by averaging over all measured jets and energy conservation requires C1(8) = 1. In QCD, 6 is proportional to the internal momentum transfer in the parton cascade and allows the determination of Ors(q2 ). The jet energy spread was measured from data taken with the MARK II detector at the electronpositron storage ring PEP at the Stanford Linear Accelerator Center. The data used in this analysis correspond to an integrated luminosity of approximately 14500 nb -1 accumulated at a center-of-mass energy of 29 GeV. The MARK II detector is composed of a large-volume solenoid magnet coaxial with the PEP beam line, a system of 16 layers of cylindrical drift chambers in the field to determine particle momenta, a set of liquid-argon-andqead shower counters outside the tracking region covering 2rr in azimuth to detect photons and identify electrons, a time-of-flight system to measure particle velocities, and a set of steel absorbers and counters to identify/a mesons. The detector has been described in detail elsewhere [4].
Events for this analysis were selected by applying the following cuts. Charged and neutral tracks had to lie in the polar angle range 50 ° < 0 < 130 ° to stay safely within the region covered by the liquid-argon shower counters. Charged tracks were required to have a minimum transverse momentum with respect to the beam axis of 100 MeV/c and photons to have a measured energy of at least 300 MeV. The particle identification capabilities of the MARK II were used to assign masses to charged particles. If the mass was ambiguous a pion mass was assumed. Photons were rejected if their distance to any charged track was less than 15 cm at the entrance of the liquid argon shower counters. All events were analysed as two-jet events. Selected events were required to have a measured thrust value greater than 0.85. This cut removed events with hard gluon radiation at large angles, and was made to justify the leading logarithm approximation [5] used in the jet calculus. The results are quite independent of the particular value of the thrust cut. The polar angle of the thrust axis had to be in the range between 65 ° and 115 ° to make sure that most of the energy flow of the jets went into the angular region where it could be measured. The measured energy of each of the two jets had to be at least 8 GeV. Each jet was required to contain at least three detected particles with at least two of them being charged. In addition the detected charged multiplicity of the event had to exceed four to discriminate against z-pair production. To remove showering Bhabha events, events were rejected if an electron with more than 8 GeV was identified. After applying the above cuts there remained 1866 jets with an average jet energy of 11 GeV.
For each opening angle 8, the total solid angle was divided into a set of calorimeters with approximately equal size. The number of calorimeters varied between 6 and 76, and the orientation of the calorimeters was chosen for each event such that the jet axis pointed into the center of a calorimeter. If E i was the energy in the ith calorimeter andMi the number of calorimeters with assigned energies different from zero, then the following moments were calculated: where N is the number of jets.
The measured values x(5) and cn(5) had to be corrected by a Monte Carlo simulation of the data for track and event selection cuts, undetected energy, initial state radiation and weak decays of charmed and bottom mesons. This correction procedure depends only on the acceptance of the detector and is insensitive to changes in the parameters of the fragmentation model. The resulting corrections for cn (6) are typically a few percent and reach 15% for larger mo- Table 1 Correction factors for the moments cn (8) and for x (8 [3] in the framework of perturbative QCD. This "jet calculus" is a probabilistic interpretation of jet development. In this picture a primary parton created in the process e+e-~ qcl at a center-ofmass energy s 1/2 develops into a parton shower by successive gluon radiation and quark-antiquark pair production. This leads to a tree-like structure where the virtual mass of the primary parton decreases successively along each branch. The shower evolution is calculated perturbatively until the virtual mass of the remaining partons are of the order of a typical hadronic mass. Then the partons turn non-perturbatively into hadrons. Since momentum transfers involved in this final hadronization process are small compared to the transverse momentum scale of the perturbative jet evolution, directional energy flow is approximately conserved. As assumed by ref. [3], these nonperturbative effects should not alter the result of the analysis, if the minimum momentum transfer observed (i.e. minimum 8) is not too small. As a result the measured hadronic energy E inside a cone of opening angle 26 originates from the decay of a virtual parton in the shower with a virtual mass up to: where the average is to be taken over all sets of calorimeters of fixed opening angles 26 and over all jets. Eq. (3) is only an upper limit for the invariant mass, since angles smaller than the size of the calorimeter cannot be resolved.
In the theory the density of such virtual partons with fractional energy x in a shower of a primary parton i with mass up to ~s 1/2 is given by a partonic fragmentation function [3,6] Di(x, s, ~ 2), (i = quark, gluon). The jet energy spread is then given by the moments of the quark fragmentation function at that 0-2: C~/(~2) =(Cxn) ` fdxx n Dq(X,s,~2). (4) q The q2 evolution of these fragmentation functions is predicted by the well known Altarelli-Parisi equations [7] which can be solved for the moments C~q with the result [8]: Here, xn, xn_ and a~, b 7 are the eigenvalues and tile first components of the corresponding eigenvectors of the matrix of anomalous dimensions as given by refs. [3,8], and b = 33 -2Nf for Nf quark flavors. The range of validity of this calculation is limited to Ots(4t ~ 2) .~ zr and ~ 2 >~ m 2 This is equivalent to hadron" the requirement that 26 must not be taken too small.
In comparing the experimental results to eq. (5), one has to choose the number of quark flavors effective in the development of the parton cascade. Recently, Edwards and Gottschalk [9] have shown that the quark mass dependent effective QCD coupling constant can be approximated sufficiently well by the formula for massless quarks if one introduces In fig. 1 we show the measurements of C2(4~ 2) and C6(4~ 2) as a function of the averaged values 4~ 2 and compare them to the predictions of eq. (5) for Nf = 3. We do not consider moments of order higher than 6 because the correction factors become large. The second-order moment C 2 is well described by eq. (5) with an a s of about 0.16 at Qo = 29 GeV even down to small values of ~ 2, where perturbative methods may not be applicable. The prediction of the moments are very sensitive to as, however the momentum transfer scale is very approximate. For the sixth order moment C 6 the agreement is still good although the best fit value of a s (29 GeV) is 0.18. The significance of the variation of a s with the order of the moments is not clear to us. Higher-order corrections to the jet calculus or residual non-perturbative effects can contribute to this difference.
Eq. (5) can be solved numerically for the ratio with as(S ) as a parameter. The agreement between data and the perturbative prediction is good for n = 2 even down to very low values of 4~ 2, where the application of the perturbative theory becomes doubtful. For n = 6 the agreement is also qualitatively as stated above but, a higher value of as(s) is required. The ratios as(4~ 2)/as(S ) derived with the assumption of 4 flavors are slightly larger and would require a value of as(S ) which is larger by a few percent. We have also compared the data to the prediction of other completely ad hoc models of e+e -~ hadrons in order to see if the jet energy moments are a sensitive discriminant among models. One simulation uses an implausible model that generates events looking nothing like the data (isotropic phase space) with the multiplicity adjusted to agree with the data. A jet axis can be determined because a finite number of particles in the final state can never give complete spherical symmetry. The moments determined from the simulation look nothing like the data in magnitude or in shape.
The second simulation generates hadrons in back-to-back jets with a transverse momentum distribution with respect to the jet axis that is gaussianly distributed and a longitudinal momentum distribution determined by phase space. Again, the mean multiplicity is adjusted to fit the real data. These events look, superficially, very much like real data, and this model as well as QCD fits the energy moments with (pl) = 400 MeV for C 2 and (p±) = 480 MeV for C 6. It is interesting to note that these values of (p±) are similar to those determined at the SPEAR storage ring for jets produced at 7.4 GeV which give (p±) = 364 -+ 2 MeV [101. Models for the jet development such as the one proposed by Feynman and Field [ 11], which are adjusted not only to fit Pi but also PU will naturally reproduce the energy moments.
In a third model we have tested the sensitivity of the jet calculus method and our experimental procedure by using a leading logarith QCD Monte Carlo [12]. The jet development in this model is determined by multiple gluon emission with a logarithmically changing coupling constant, as ~ 1/ln(q2/A2) • Since A is a parameter, we were able to examine the sensitivity of the experimental procedure to a variation of a s.
In conclusion, this analysis shows that the perturbatire QCD jet calculus gives a good description of the jet energy moments. In the framework of this model we have extracted as at different momentum transfers and we have demonstrated that the data require a decreasing value of as with increasing energy. This meth-od alone is not sufficient to distinguish QCD from simple limited transverse momentum models.
We wish to acknowledge stimulating discussions with our theoretical collegues S. Drell, S. Sharpes, L. Trentadue and P. Zerwas.