Investigating quark and gluon fragmentation

Volume 168B, number 1,2

PHYSICS LETTERS

27 February 1986

INVESTIGATING QUARK AND GLUON FRAGMENTATION C.J. M A X W E L L Department of PhvsR~, Unir,ersi(v of Durham, Durham DHI 3I.E. UK Received 27 June 1985; revised manuscript received 11 September 1985

We propose as jet fragmentation variables the relative energies of successive energy-ordered charged hadrons. The jet fragmentation function in terms of these variables is insensitive to the algorithm used to define the jets. and differs significantly for light quark (u. d, s) and gluon jets. Such variables can provide experimentally useful jet measures.

An ability to distinguish between quark and gluon jets would considerably extend the range of experimental tests of perturbative QCD. Obviously, such a distinction cannot be made on an event-by-event basis, but by constructing suitable jet measures [1], which are then averaged over many events, one could hope to obtain information on the relative fractions of quark and gluon jets in a data sample. The angular distribution and separate production cross sections for quark and gluon jets could also be investigated. In this letter we wish to investigate which hadronic distributions one should consider in order to best construct experimentally useful measures of this kind. The features we shall require of a hadronic distribution are: (i) The distribution should be (approximately) independent of the jet energy, which is poorly determined by experhnent. (ii) The distribution should be insensitive to the details of the algorithm used to define the jet, and to experimental ambiguities such as missing soft particles. (iii) The distribution should differ significantly for light quark and gluon jets. An obvious candidate distribution is the inclusive charged hadron fragmentation function for parton type i, D i ( z ) , where z = Eh//~je t is the fraction of the jet energy carried by the hadron. This distribution satisfies (i) and (iii) but does not satisfy (ii). This is because Eje t is sensitive to missing soft particles at low z, of which a gluon jet possesses more than a quark. This means that z is experimentally overestimated for 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

a gluon jet somewhat more than for a quark jet. Consequently the measured differences between the two fragmentation functions are greatly reduced. Poor measurement of jet energy has proved a problem in investigations of fragmentation at the ~p collider where the initial results suggested no significant difference between quark and gluon fragmentation [2]. A reanalysis involving an elaborate correction procedure for the jet energy has revealed differences [3]. The procedure introduces large systematic errors and requires a detailed computer modeling of tile detector response. If one could find distributions sarisfying (i)-(iii) they would be of use in more flexible, less Monte Carlo dependent fragmentation tests. To search for such distributions we have used Webber's jet Monte Carlo computer program BIGWIG [4] to generate large samples of qTq (q denoting a light quark u, d, s) and gg two-jet events at CM energies x/~ = 50, 100,200 GeV ( ~ 5 0 000 events of each parton type at each energy). We shall not consider heavy flavour distributions in this paper. Sukhatme [5] has previously discussed the use of multihadron fragmentation functions D i ( X l , x 2 , x 3 , .... X N ; h 1 , h 2 , ..., hN) in jet identification, D i denothag the probability density for parton type i to fragment to hadrons hi, h2, ..., hN, having momentum fractions x 1, x2, .--, X N . Since, as we have noted above, the jet energy (momentum) cannot be experimentally determined reliably, we consider instead ordering the charged hadrons in energy -Eha > E b 2 > Eh3 .... and then constructing the dimensionless 131

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variables x21 = Eh2/Eh, , x32 = Eh3/Eh2 , x43 = Eh4/Eh3, etc. One can then consider the fragmentation function D[(x21 , x32, x43 .... ) (we shall be inclusive with respect to hadron type for the moment). Our Monte Carlo studies reveal that for the most energetic four charged particles in a jet (four being chosen purely because of limited statistics) there is an interesting factorisation in these variables for light quarks and gluons:

Di(x21 , x32 , x43 ) -~ D2i(x21 )D3~x32)Dai(X43 ) . (1)

(a) 2

r--I i i

D2( X21) --'4

'15

i i I

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The successive relative energies are uncorrelated within statistical errors. The factorized fragmentation histograms D2(x), D3(x ), D4(x ) are plotted in figs. I a - 1 c. These distributions do not differ significantly between u, d, s flavours and are essentially independent of CM energy x/~, and so we plot a single solid-line "u, d, s" distribution which is in fact that for u quarks at X,ff = 100 GeV, and a dashed-line distribution for gluon jets at the same energy. We also checked that ordering in momentum rather than energy and removing a thrust cut applied to the two-jet events did not significantly change the distributions either. Lack o f correlation in these variables between the two jets in each event was also checked. These distributions certainly satisfy our requirements (i)-(iii). We shall quantify in a moment the significance of the cL/g difference, but it is certainly clearly visible, the gluon distribution being more concentrated at large x than that o f the quark. First we wish to discuss briefly the physical origin of the factorization. To help clarify this we constructed a simple one-dimensional recursive cascade Monte "4

-05

(C)

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2

3

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5

•6

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9

X21

"25

3 (b) ! I I I r--4

J I I I I

I

I

25 D~(x32) 2

2

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-1

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Fig. 1. (a), Co), (c), the fragmentation functions D2 (x21), D3(x32),D4(x43 ) (see text). The solid-line distribution is for light quark jets (u, d, s) and the dashed-line distr~ution for gluonjets.

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Carlo computer program involving "quarks" and "mesons" in which the branching q(p) ~ q((l - x ) p ) + M(xp), occurs with the momentum fraction x distributed according to a momentum sharing function f ( x ) d x (we chose 2(1 - x)dx, which is a popular choice in such models [5]). We generated 106 events and noted an approximate factorization (statisticallysignificant deviations were observable with these large statistics). We conclude that such a factorization is an approximate property of a Markov-type recursive cascade. We have not been able to provide an analytical justification for this result. Since recursive QCD partonic branching underlies the BIGWIG Monte Carlo this is presumably the physical origin o f the effect. The more pronounced peaking of D2, D3, D 4 at large x for gluon jets compared with quarks is a necessary consequence of the softer, higher multiplicity, gluon fragmentation expected from QCD. The increasing peaking of the successive D2, D3, D 4 distributions at large x values is also reproduced in the simple recursive cascade study. To quantify the g/q differences in the distributions o f fig. 1 we shall apply the " o p t i m u m " j e t measures introduced in ref. [1 ]. Given some measurable jet property V one can construct jet measures Q, G,

v-

v - q

Q = (V)q - (V)g '

G = (V)g

(V)q .

(2)

We can see on inspection from eq. (2) that (Q)q = 1, (Q)g = 0 and similarly (G)q = 0, (G)g = 1. Q and G consequently have the property that when averaged over a mixture o f quark and gluon jets, J, (Q)j and (G)j estimate respectively the fraction o f quarks and gluons in the mixture,fq,fg. If Vis chosen to be the conditional probability that a jet with hadronic configuration H (the jet propertieswe choose to measure) is the result of q fragmentation, these measures estimate fq, fg with the least possible variance (same as a maximum likelihood fit [6]). The reader is urged to consult refs. [1,6] for full details. The minimized variance is [ 1,6] [ r

(p(q-~H)_p(g-~H)2 \-1

varQ=~JdH f q ~ ~ ~ ) ]

,

(3)

where p(i ~ H) denote the fragmentation probabilities for parton type i. If we use a discrete choice for H, e.g. the most energetic hadron types h 1 = r¢+, h 2 = K - , then f dH should be replaced by Z H.

27 February 1986

Table 1 The standard deviation c in a measurement of fq, fg u~ng the optimum jet measure Q, for various ehoiees of jet hadronic conf'~uratio n H. H

c

x2t

2.8

x32

2.5

x43

2.4

x21, x32

1.9

X21,X32,X43 h I , h2 x21, hi, h2

1.6 5.3 2.2

In table 1 we indicate, for various choices o f H, the coefficient c (= x/var-'~-~ o f 1/x/TV in the error on a determination offq(fg) from a sample o f N jets, as obtained from our Monte Carlo runs. The smaller c is, the bigger the difference between q and g fragmentation. Notice that c ": 1 for perfect event-by-event classification. The variance of eq. (3) involves the fractions f q , f g - we have determined c in table 1 for a 5 0 : 5 0 q : g mixture. We see that using the most energetic hadron types alone to classify the jet i.e. h 1 = n +, h 2 = K - etc., does not provide good discrimination (c = 5.3). The Monte Carlo results reveal that Di(x21; h 1 , h 2) "" D2i(x21)Dih(h I , h 2 ) ,

(14)

where Dih is rather similar for quarks and gluons. Dull, Ddh, Dsh do show some differences. Table 1 indicates that using the xi/variables of the most energetic three or four particles in a jet can enablefq (fg) to be measured with reasonable accuracy (+ 0.06 for 1000 jets using the most energetic three charged hadrons). As a test we calibrated the jet measure Q with 10000 ufi and gg events at x/s-= 100 GeV using the most energetic three charged hadrons to determine x21, x32 (we used bins in x of width 0.25 and hence 16 possible jet configurations). We then used the measure so constructed to recover fq,fg for various q/g mixtures (q being itself a mixture of u, d, s) at x/~ = 50 GeV and x/s = 200 GeV. We found a <~ 20% systematic shift in the mean fq and fg extracted at the higher and lower energies relative to the calibration energy x/~ = 100 GeV, due to the small but non-zero scaling violations o f the x21 , x32 distributions. 133

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Such jet measures should be useful in a wide variety of jet analyses. It is clearly important to experimentally measure the D2, D3, D 4 structure functions and check the factorization o f eq. (1). In large-p T collider twojet events, the fraction o f quark/gluon jets in a jet sample is simply related to averages o f proton structure function ratios over the events o f the sample (this is a consequence o f the "single effective subprocess" approximation of ref. [ 1 ]). Finally, the insensitivity o f these xi/hadronic distributions to details of jet definition, and to the process in which the jets are produced, should also prove useful in confirming whether or not coUider monojets and other exotica involve "conventional" QCD jets [7]. We wish to thank Bryan Webber for making BIGWIG available to the Durham Theory Group, and Mike Whalley for crucial help with various computing

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problems. We are grateful to Alan Martin for reading the manuscript and some useful comments.

References [ 1 ] B.L. Combridge and CJ. Maxwell, Nucl. Phys. B239 (1984) 429. [21 UA1 Collab., G. Arnison et al., Phys. Lett. 132B (1983) 223. [3] P.Ghez,Proc. p~ CoUiderWorkshop (Aosta, Italy, 1985). [4] G. Maxchesini and B.R. Webber, Nucl. Phys. B238 (1984) 1; B.R. Webber, Nucl. Phys. B238 (1984) 492. [5 ] U.P. Sukhatme, Phys. Lett. 113B (1982) 185 ; Proe. of the Europhysics Conf. on Patrons in soft hadronic processes (Erice, Sicily 1981), ed. R. Van de WaUe. [6] B.L. Combridge and CJ. Maxwell, Durham preprint, in preparation. [7] S.D. EUis, R. Kleiss and WJ. Stiffing, CERN preprint TH.4170/85 (1985).