Nuclear Instruments and Methods in Physics Research A 495 (2002) 107–120 On the energy measurement of hadron jets Olga Lobban, Aravindhan Sriharan, Richard Wigmans* Department of Physics, Texas TECH University, Box 41051, Lubbock, TX 79409-1051, USA Received 16 July 2002; received in revised form 26 August 2002; accepted 28 August 2002 Abstract The elementary constituents of hadronic matter (quarks, anti-quarks, gluons) manifest themselves experimentally in the form of jets of particles. We investigate the precision with which the energy of these fragmenting objects can be measured. The relative importance of the instrumental measurement precision and of the jet algorithm is assessed. We also evaluate the ‘‘energy flow’’ method, in which the information from a charged-particle tracker is combined with that from a calorimeter in order to improve the jet energy resolution. r 2002 Published by Elsevier Science B.V. PACS: 02.70.Uu; 29.40.Vj Keywords: Calorimetry; Fluctuations; Jets; Energy flow 1. Introduction detector quality. Usually, a jet is defined as the collection of particles that fall within a cone with Matter as we know it consists of leptons and opening angle R emerging from the interaction quarks. Whereas the properties of leptons such as vertex. Typical values of R ; when expressed in electrons or muons can usually be measured with a terms of an interval in Z ; f space q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi very high degree of precision, the same is not true D Z 2 þ D f 2 ( R ¼ ), range from 0.3 to 0.7. If the for quarks. Quarks are ‘‘locked up’’ inside mesons chosen R value is large, the cone may be or (anti-)baryons and any attempt to isolate them contaminated with particles that have nothing to creates more such particles. In high-energy scatter- do with the fragmenting object, if R is small, some ing experiments aimed at studying their properties, jet fragments may be located outside the cone. quarks, diquarks or anti-quarks fragment into jets Fluctuations in the jet energy contained within the of hadrons. jet-defining cone form an irreducible component of The precision with which the properties of the the jet energy resolution. fragmenting object can be measured depends on At energies below 100 GeV ; the contributions of two factors: The jet-defining algorithm and the this irreducible component are substantial and in practical experiments they are the main factor limiting the jet energy resolution. However, at *Corresponding author. Tel.: +1-806-742-3779; fax: +1- higher energies, jets become more and more 806-742-1182. collimated and the effects of the jet algorithm on E-mail address: wigmans@ttu.edu (R. Wigmans). 0168-9002/02/$ - see front matter r 2002 Published by Elsevier Science B.V. PII: S 0 1 6 8 - 9 0 0 2 ( 0 2 ) 0 1 6 1 5 - 7
108 O. Lobban et al. / Nuclear Instruments and Methods in Physics Research A 495 (2002) 107–120 the energy resolution diminish correspondingly. In representation of the physics processes taking Section 2 of this paper, we investigate the energy place in practice. However, it does contain the dependence of these effects. essential elements necessary to evaluate the energy One of the problems in designing calorimeter dependence of the contributions of the systems for modern experiments is the fact that the jet algorithm to the resolution. In our program, requirements for excellent energy resolution for the fragmentation process is governed by a single hadrons and jets are orthogonal to those for fragmentation function high-resolution electromagnetic (em) calorimetry D ð z Þ ¼ ð a þ 1 Þ ð 1 � z Þ a ð 1 Þ [1]. High-resolution hadronic shower measure- z ments require compensating calorimeters. And in which D ð z Þ denotes the probability that a jet compensation (i.e. equal calorimeter response to fragment carries a fraction z of the energy of the the em and non-em components of hadron fragmenting object [4]. The parameter a can be showers, e = h ¼ 1 : 0) is only achieved in sampling chosen as desired. It has been demonstrated that a calorimeters with a very small sampling fraction, function of this type gives a reasonable description e.g., 2.3% in lead/plastic-scintillator structures. On of the fragmentation processes measured at LEP the other hand, high-resolution em shower detec- and at the Tevatron, for parameter values a ¼ 3 tion requires an instrument with a large sampling and 6, respectively [5]. fraction, e.g., 100% in crystals or > 40 % in In our Monte Carlo program, jet fragments are detectors such as the NA48 LKr calorimeter [2]. generated with energies zE jet ; with the values of z In order to solve this dilemma, it has been chosen from a distribution representing Eq. (1). proposed that one could significantly improve the Each fragment is assigned a mass m ; a charge and performance of a poor-resolution hadronic calori- a transverse momentum p > : Ten percent of the meter system by combining its information with particles are assumed to be kaons and ninety that of an upstream tracker system. In this percent pions. One-third of the particles are approach, sometimes referred to as the Energy electrically neutral, the rest are charged. The Flow Method , the momenta of the charged jet transverse momentum is chosen from an exponen- fragments measured with high precision by the tially falling distribution with a mean value of tracker serve as a first-order estimate of the jet 0 : 3 GeV = c : If the chosen parameters yield an energy. Second-order corrections, intended to unphysical result, e.g., if the chosen mass is larger account for the neutral jet component, are derived than the fragment’s energy zE jet ; or if the from the calorimeter signals. Of course, the transverse momentum is larger than the total contributions of showering charged particles to q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð zE jet Þ 2 � m 2 the calorimeter signals have to be discounted momentum ; the fragment is dis- properly for this method to work. Methods of carded and a new one is selected. The selection of this type have been successfully used to improve jet fragments is continued until the jet energy is the resolution of jets from Z-decay at LEP [3]. In exceeded. In that case, the energy of the last Section 3 of this paper, we investigate the fragment is reduced so that the total energy of all prospects of such methods at higher energies. fragments combined equals the jet energy. Concluding remarks are given in Section 4. We used this program to generate jets with fixed energies, ranging from 10 to 1000 GeV : For each energy, 10 000 jets were generated for two different 2. Effects of the jet algorithm values of the fragmentation function parameter: a ¼ 3 and a ¼ 6 : First, we show some general We have studied the effect of a jet-defining results that give an impression of the character- algorithm on the energy resolution for fragment- istics of the generated jets. Fig. 1 shows the energy ing quarks with a Monte Carlo program that distribution for the particles that constitute a we developed for this purpose. This Monte 100 GeV jet, fragmenting according to a ¼ 3 or Carlo program is based on a highly simplified a ¼ 6 : In Fig. 2, the distribution of the jet
O. Lobban et al. / Nuclear Instruments and Methods in Physics Research A 495 (2002) 107–120 109 Fig. 3. The average fragment multiplicity and the average Fig. 1. Energy distribution of particles generated in the number of fragments that is minimally needed to account for fragmentation of a 100 GeV jet according to Eq. (1), with a ¼ 90% of the jet’s energy as a function of the jet energy, for two 3 and 6, respectively. different values of the fragmentation function parameter a : a ¼ 3 jets the 10 most energetic particles carry 90% of the total jet energy. For a ¼ 6 jets, that takes 15 particles, on average. This is true at all energies, which is of course a direct consequence of the very concept of a fragmentation function that depends only on z : We defined the cone parameter R that formed the basis of the applied jet algorithm as q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð D f Þ 2 þ ð D Z Þ 2 R ¼ ð 2 Þ where D f and D Z denote the spread around the nominal direction of the fragmenting object in the azimuthal and polar angles, respectively. The fate of a jet fragment was decided on the basis of the ratio of its transverse and longitudinal Fig. 2. Multiplicity distribution of fragments from 100 GeV jets, fragmenting according to Eq. (1), with a ¼ 3 and a ¼ 6 ; momenta, p > = p jj : If respectively. arctan ð p > = p jj Þ > R = 2 fragment multiplicity is given for 100 GeV jets and then the fragment fell outside the cone, otherwise the average multiplicity is shown as a function of it was considered to contribute to the measured the jet energy in Fig. 3. jet characteristics (energy, momentum, composi- In spite of the large numbers of particles tion). We thus implicitly ignored the effects of constituting the jets, only relatively few particles an eventual magnetic field, which has the tendency contribute substantially to the total energy. to sweep soft charged particles out of the This is also shown in Fig. 3. For example, in cone. Therefore, the results to be presented are
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