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arXiv:hep-ph/0312283v2 2 Jan 2004

CERN-TH/2003-306

LPTHE-03-40

hep-ph/0312283

Event shapes in e+e−annihilation and deep inelastic

scattering

Mrinal Dasgupta† and Gavin P. Salam‡

† CERN, Theory Division, CH-1211 Geneva 23, Switzerland.

‡ LPTHE, Universities of Paris VI and VII, and CNRS UMR 7589, Paris 75005,

France.

Abstract.

annihilation and DIS. It includes discussions of perturbative calculations, of

various approaches to modelling hadronisation and of comparisons to data.

This article reviews the status of event-shape studies in e+e−

1. Introduction

Event shape variables are perhaps the most popular observables for testing QCD and

for improving our understanding of its dynamics. Event shape studies began in earnest

towards the late seventies as a simple quantitative method to understand the nature

of gluon bremsstrahlung [1–5]. For instance, it was on the basis of comparisons to

event shape data that one could first deduce that gluons were vector particles, since

theoretical predictions employing scalar gluons did not agree with experiment [6].

Quite generally, event shapes parametrise the geometrical properties of the

energy-momentum flow of an event and their values are therefore directly related to

the appearance of an event in the detector. In other words the value of a given event

shape encodes in a continuous fashion, for example, the transition from pencil-like

two-jet events with hadron flow prominently distributed along some axis, to planar

three-jet events or events with a spherical distribution of hadron momenta. Thus they

provide more detailed information on the final state geometry than say a jet finding

algorithm which would always classify an event as having a certain finite number of

jets, even if the actual energy flow is uniformly distributed in the detector and there

is no prominent jet structure present in the first place.

Event shapes are well suited to testing QCD mainly because, by construction, they

are collinear and infrared safe observables. This means that one can safely compute

them in perturbation theory and use the predictions as a means of extracting the strong

coupling αs. They are also well suited for determinations of other parameters of the

theory such as constraining the quark and gluon colour factors as well as the QCD beta

function. Additionally they have also been used in other studies for characterising the

final state, such as investigations of jet and heavy quark multiplicities as functions of

event shape variables [7,8].

Aside from testing the basic properties of QCD, event shape distributions are a

powerful probe of our more detailed knowledge of QCD dynamics. All the commonly

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Event shapes in e+e−annihilation and deep inelastic scattering

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studied event shape variables have the property that in the region where the event

shape value is small, one is sensitive primarily to gluon emission that is soft compared

to the hard scale of the event and/or collinear to one of the hard partons. Such small-

transverse-momentum emissions have relatively large emission probabilities (compared

to their high transverse momentum counterparts) due to logarithmic soft-collinear

dynamical enhancements as well as the larger value of their coupling to the hard

partons. Predictions for event shape distributions therefore typically contain large

logarithms in the region where the event-shape is small, which are a reflection of the

importance of multiple soft and/or collinear emission. A successful prediction for an

event shape in this region requires all-order resummed perturbative predictions or

the use of a Monte-Carlo event generator, which contains correctly the appropriate

dynamics governing multiple particle production. Comparisons of these predictions to

experimental data are therefore a stringent test of the understanding QCD dynamics

that has been reached so far.

One other feature of event shapes, that at first appeared as an obstacle to their

use in extracting the fundamental parameters of QCD, is the presence of significant

non-perturbative effects in the form of power corrections that vary as an inverse

power of the hard scale, (Λ/Q)2p.For most event shapes, phenomenologically it

was found that p = 1/2 (as discussed in [9]), and the resulting non-perturbative

effects can be of comparable size to next-to-leading order perturbative predictions,

as we shall discuss presently.The problem however, can be handled to a large

extent by hadronisation models embedded in Monte-Carlo event generators [10–12],

which model the conversion of partons into hadrons at the cost of introducing several

parameters that need to be tuned to the data. Once this is done, application of such

hadronisation models leads to very successful comparisons of perturbative predictions

with experimental data with, for example, values of αsconsistent to those obtained

from other methods and with relatively small errors.

However since the mid-nineties attempts have also been made to obtain a

better insight into the physics of hadronisation and power corrections in particular.

Theoretical models such as those based on renormalons have been developed to probe

the non-perturbative domain that gives rise to power corrections (for a review, see

[13]). Since the power corrections are relatively large effects for event shape variables,

scaling typically as 1/Q (no other class of observable shows such large effects), event

shapes have become the most widely used means of investigating the validity of these

ideas. In fact an entire phenomenology of event shapes has developed which is based

on accounting for non-perturbative effects via such theoretically inspired models, and

including them in fits to event shape data alongside the extraction of other standard

QCD parameters. In this way event shapes also serve as a tool for understanding more

quantitatively the role of confinement effects.

The layout of this article is as follows.

definitions of several commonly studied event shapes in e+e−annihilation and deep

inelastic scattering (DIS) and discuss briefly some of their properties. Then in section 3

we review the state of the art for perturbative predictions of event-shape mean values

and distributions. In particular, we examine the need for resummed predictions for

distributions and clarify the nomenclature and notation used in that context, as

well as the problem of matching these predictions to fixed order computations. We

then turn, in sections 4 and 5, to comparisons with experimental data and the issue

of non-perturbative corrections, required in order to be able to apply parton level

calculations to hadronic final state data. We discuss the various approaches used to

In the following section we list the

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estimate non-perturbative corrections, ranging from phenomenological hadronisation

models to analytical approaches based on renormalons, and shape-functions.

also discuss methods where the standard perturbative results are modified by use

of renormalisation group improvements or the use of ‘dressed gluons’. We study fits

to event shape data, from different sources, in both e+e−annihilation and DIS and

compare the various approaches that are adopted to make theoretical predictions. In

addition we display some of the results obtained by fitting data for various event shape

variables for parameters such as the strong coupling, the QCD colour factors and the

QCD beta function. Lastly, in section 6, we present an outlook on possible future

developments in the field.

We

2. Definitions and properties

We list here the most widely studied event shapes, concentrating on those that have

received significant experimental and theoretical attention.

2.1. e+e−

The canonical event shape is the thrust [2,14], T:

T = max

? nT

?

i|? pi.? nT|

?

i|? pi|

, (1)

where the numerator is maximised over directions of the unit vector ? nT and the sum

is over all final-state hadron momenta pi(whose three-vectors are ? piand energies Ei).

The resulting ? nT is known as the thrust axis. In the limit of two narrow back-to-back

jets T → 1, while its minimum value of 1/2 corresponds to events with a uniform

distribution of momentum flow in all directions. The infrared and collinear safety

of the thrust (and other event shapes) is an essential consequence of its linearity in

momenta. Often it is 1 − T (also called τ) that is referred to insofar as it is this that

vanishes in the 2-jet limit.

A number of other commonly studied event shapes are constructed employing

the thrust axis. Amongst these are the invariant squared jet-mass [15] and the jet-

broadening variables [16], defined respectively as

ρℓ,r =

??

?

i∈Hℓ,rpi

(?

2?

?2

iEi)2

, (2)

Bℓ,r=

i∈Hℓ,r|? pi×? nT|

i|? pi|

, (3)

where the plane perpendicular to the thrust axis† is used to separate the event into left

and right hemisphere, Hℓand Hr. Given these definitions one can study the heavy-jet

mass, ρH = max(ρℓ,ρr) and wide-jet broadening, BW = max(Bℓ,Br); analogously

one defines the light-jet mass (ρL) and narrow-jet broadening BN; finally one defines

also the sum of jet masses ρS= ρℓ+ρrand the total jet broadening BT = Bℓ+Brand

their differences ρD= ρH− ρL, and BD= BW− BN. Like 1 − T, all these variables

(and those that follow) vanish in the two-jet limit; ρLand BNare special in that they

also vanish in the limit of three narrow jets (three final-state partons; or in general

† In the original definition [15], the plane was chosen so as to minimise the heavy-jet mass.

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for events with any number of jets in the heavy hemisphere, but only one in the light

hemisphere).

Another set of observables [17] making use of the thrust axis starts with the thrust

major TM,

?

where the maximisation is performed over all directions of the unit vector ? nM, such

that ? nM.? nT= 0. The thrust minor, Tm, is given by

?

and is sometimes also known [18] as the (normalised) out-of-plane momentum Kout.

Like ρLand BN, it vanishes in the three-jet limit (and, in general, for planar events).

Finally from TM and Tmone constructs the oblateness, O = TM− Tm.

An alternative axis is used for the spherocity S [2],

TM= max

? nM

i|? pi.? nM|

?

i|? pi|

,? nM.? nT = 0, (4)

Tm=

i|? pi.? nm|

?

i|? pi|

,? nm= ? nT×? nM,(5)

S =

?4

π

?2

max

? nS

??

i|? pi×? nS|

?

i|? pi|

?2

. (6)

While there exist reliable (though algorithmically slow) methods of determining the

thrust and thrust minor axes, the general properties of the spherocity axis are less well

understood and consequently the spherocity has received less theoretical attention. An

observable similar to the thrust minor (in that it measures out-of-plane momentum),

but defined in a manner analogous to the spherocity is the acoplanarity [19], which

minimises a (squared) projection perpendicular to a plane.

It is also possible to define event shapes without reference to an explicit axis. The

best known examples are the C and D parameters [20] which are obtained from the

momentum tensor [21]‡

??

where paiis the athcomponent of the three vector ? pi. In terms of the eigenvalues λ1,

λ2and λ3of Θab, the C and D parameters are given by

Θab=

i

paipbi

|? pi|

?

/

?

i

|? pi|, (7)

C = 3(λ1λ2+ λ2λ3+ λ3λ1),D = 27λ1λ2λ3.(8)

The C-parameter is related also to one of a series of Fox-Wolfram observables [4], H2,

and is sometimes equivalently written as

C =3

2

?

i,j|? pi||? pj|sin2θij

(?

i|? pi|)2

. (9)

The D-parameter, like the thrust minor, vanishes for all final-states with up to 3

particles, and in general, for planar events.

‡ We note that there is a set of earlier observables, the sphericity [22], planarity and aplanarity [23]

based on a tensor Θab=?

ipaipbi. Because of the quadratic dependence on particle momenta, these

observable are collinear unsafe and so no longer widely studied.

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The C-parameter can actually be considered (in the limit of all particles

being massless) as the integral of a more differential observable, the Energy-Energy

Correlation (EEC) [5],

dΣ(χ)

dcosχ

?

(?

where the average in eq. (10) is carried out over all events.

Another set of variables that characterise the shape of the final state are the n-jet

resolution parameters that are generated by jet finding algorithms. Examples of these

are the JADE [24] and Durham [25] jet clustering algorithms. One introduces distance

measures yij

=2EiEj(1 − cosθij)

(?

y(Durham)

ij

(?

for each pair of particles i and j. The pair with the smallest yij is clustered (by

adding the four-momenta — the E recombination scheme) and replaced with a single

pseudo-particle; the yijare recalculated and the combination procedure repeated until

all remaining yijare larger than some value ycut. The event-shapes based on these jet

algorithms are y3≡ y23, defined as the maximum value of ycutfor which the event is

clustered to 3 jets; analogously one can define y4, and so forth. Other clustering jet

algorithms exist. Most differ essentially in the definition of the distance measure yij

and the recombination procedure. For example, there are E0, P and P0 variants [24]

of the JADE algorithm, which differ in the details of the treatment of the difference

between energy and the modulus of the 3-momentum. The Geneva algorithm [26] is

like the JADE algorithm except that in the definition of the yij it is Ei+ Ej that

appears in the denominator instead of the total energy. An algorithm that has been

developed and adopted recently is the Cambridge algorithm [27], which uses the same

distance measure as the Durham algorithm, but with a different clustering sequence.

We note that a number of variants of the above observables have recently been

introduced [28], which differ in their treatment of massive particles. These include the

p-scheme where all occurrences of Eiare replaced by |? pi| and the E-scheme where each

3-momentum ? piis rescaled by Ei/|? pi|. The former leads to a difference between the

total energy in the initial and final states, while the latter leads to a final-state with

potentially non-zero overall 3-momentum. While such schemes do have these small

drawbacks, for certain observables, notably the jet masses, which are quite sensitive to

the masses of the hadrons (seldom identified in experimental event-shape studies), they

tend to be both theoretically and experimentally cleaner. Ref. [28] also introduced a

‘decay’ scheme (an alternative is given in [29]) where all hadrons are artificially decayed

to massless particles. Since a decay is by definition a stochastic process, this does not

give a unique result on an event-by-event basis, but should rather be understood as

providing a correction factor which is to be averaged over a large ensemble of events.

It is to be kept in mind that these different schemes all lead to identical perturbative

predictions (with massless quarks) and differ only at the non-perturbative level.

A final question relating to event-shape definitions concerns the hadron level at

which measurements are made. Since shorter lived hadrons decay during the time of

=

dσ

σdcosχ= ?EEC(χ)?

i,jEiEjδ(cosχ − cosθij)

iEi)2

(10)

EEC(χ) = (11)

y(Jade)

ij

kEk)2

i,E2

, (12)

=2min(E2

j)(1 − cosθij)

kEk)2

, (13)