F.V.Tkachov Phys. Rev. Lett. 73 (1994) 2405; Erratum 74 (1995) 2618 1


Measuring the number of hadronic jets

F.V. Tkachov*(a)
Physics Department, Penn State University, State College, Pennsylvania 16802

(Received XX Xxxxxx 1994)

A quantitative description of the qualitative feature of multihadron final states known as the
"number of jets" is given by a sequence of infrared finite shape observables (jet discriminators) that:
take continuous values between 0 and 1; are stable--unlike clustering algorithms--against small
variations of the input (data errors, Sudakov effects etc.); have a form of multiparticle correlators
that is natural in the context of quantum field theory and hence are better suited for a systematic
study of theoretical uncertainties (logarithmic and power corrections).


1. The jet paradigm is the foundation of the high-energy On the measurement side, any algorithm that produces an
collider physics1. It is based on the experimental evidence for integer number of jets cannot be fully satisfactory-- even be-
hadronic jets2 and the Quantum Chromodynamics-based pic- fore any dynamics gets involved. Indeed, such an algorithm
ture of hadronic energy flow inheriting the shape of partonic rips the continuum of multiparticle states by mapping it to the
energy flow in the underlying hard process3. However, the discrete set of natural numbers. A discontinuous mapping is
problem of adequate numerical description of multijet structure unstable with respect to small variations (measurement errors
of multihadron events proved theoretically subtle, its apparent or unknown higher order corrections) for some values of input
simplicity turned out deceptive, while its satisfactory solution, data (cf. Fig. 1).11 As a result, the inversion of hadronization is
elusive. The fundamental role of calorimetric measurements in a mathematically ill-posed problem, whence the problem of
high-energy experiments warrants a scrutiny of the logical spurious jets, and the sensitivity to Sudakov effects and to irre-
principles of such measurements. levant details of recombination procedures.8 This pathology is
2. It makes sense to divide the problems where jets are somewhat masked off by averaging over many events. But a
studied into two classes. The descriptive theory of hadronic deterministic recombination algorithm is applied separately to
jets studies the dynamics of jets as such4; one is mostly inter- each stochastically generated event. So, instead of a statistical
ested in qualitative effects that occur in the leading logarithmic compensation of errors, there occurs a smearing between cross
order; a systematic improvement of theoretical predictions is, sections with adjacent numbers of jets.12 It can be eliminated
typically, hardly possible.5 neither by increasing statistics, nor by varying the jet resolution
The second class (precision measurements) comprises ycut , and it is more important for smaller ycut , lower energies
quantitative studies of the Standard Model (determination of and larger numbers of jets.13
2
S (Q ) etc.1) where one aims at a highest reliability for What, then, could be a quantitative measure for the qualita-
both data and theoretical predictions: tive feature of multiparticle final states known as the "number
Reliability of data means that the problem ought to be re- of jets", a measure that allows a correct handling of data errors
garded as the one of measurement rather than the one of mod- and a systematic study of theoretical uncertainties, and that is
eling dynamics. One has to ensure that measurements be stable unbiased by the imperfect knowledge of jet dynamics?
with respect to errors in data from calorimeter cells, their posi- 4. MATHEMATICAL NATURE OF "ENERGY FLOW". If is a calo-
tion and geometry, etc. (otherwise physical information may be rimeter cell, then the energy deposited in it by particles that hit
distorted by artefacts of measurement procedures), and that the the cell is E( ) 0 . Energy conservation implies that if one
data experimentalists produce be not biased by the imperfect takes two non-overlapping cells and ' and combines them
knowledge of details of dynamics. into one, then the energy deposited in it is the sum of energies
Reliability of theoretical predictions means that it ought to deposited in and ' separately: E( ) = E( ) + E( ) .
be possible to systematically include logarithmic and power One can consider cells simply as parts (subsets) of the unit
corrections. The observables one uses ought to conform to the sphere around the collision point. Then the energy flow (EF for
 17 Jan 1999 general structure of the underlying formalism (perturbative short) is a non-negative additive function on the subsets .
Quantum Field Theory) to ensure a better control over theore-
tical uncertainties due to a considerable sophistication of the Such functions are known as abstract measures.14, 15
modern analytical methods of the theory of Feynman diagrams6. Let P be a multiparticle state, P = {E , $p }
i i i , where Ei and

3. Jet counting is an attempt to use jets of hadrons to tag $
pi are the energy and direction (a unit 3-vector) of the i-th parti-
events. Its great usefulness7 is due to the fact that the very cle. All information about P obtainable using calorimeters is its EF
presence of jets and their number is the most direct and clear represented as a linear combination of -functions localized at $pi ,
manifestation of the dynamics of QCD. =
The conventional jet counting determines an integer number EP ( $p) E $ ( $ )
p , (1)
i i pi
of jets for each event using algorithms8 which attempt to recon- where $p is a variable unit 3-vector running over the sphere.
struct the underlying partons' momenta by, in effect, inverting The energy measured by a cell is
the hadronization. They were invented9 in the context of de-
scriptive theory of jets, involve many ambiguities8, and their E ( ) $
= dp E ( $p) = E .
P z
P $
p
i
i
use in measurement-type problems may not be accepted un- The observables we deal with in calorimetric measurements
critically: are functions of EFs E. Let f (E) be such a function. Its stabi-
On the theory side, the definition of jets in such algorithms uses
phase space cutoffs to take into account cancellations of IR singula- lity with respect to data errors translates into a concrete kind of
rities. This is rather unnatural within the formalism of QFT: one continuity. Let En be a sequence of EFs such that, however
has to recur to numerics even in simpler cases10 whereas a study small the energy resolution and geometry of calorimeters, En
of power corrections remains practically impossible. become indistinguishable within data errors for all n large


F.V.Tkachov Phys. Rev. Lett. 73 (1994) 2405; Erratum 74 (1995) 2618 2


r
enough. This calorimetric or C-convergence of EFs is for- where p 0 = = $
i are light-like 4-momenta ( p E
i i , p E p
i i i ) and
malized as follows. Let 0 ( $p) 1 be a continuous function the 4-vector P2 = 1 describes the reference frame22. Symmetry
on the unit sphere. It can be thought of as describing local effi- yields a similar factor for each pair of arguments. One gets the
ciency of a calorimeter cell: for a given EF E( $p) the ex- sequence of jet discriminators:
pression z d$p E( $p)( $)
p E is the energy measured by this J (E , p$ ,K, E , p$ ) = E K E j ( p$ ,K, p$ )
,
m 1 1 N N i i m i i
cell. Then C-convergence of E 1 m 1 m
n is equivalent to numerical 1i K
< <i N
1 m
convergence of E for any "detector" .16 For a correctly
n
j ( p$ ,K, p$ ) = N . (5)
defined observable f , f ( m 1 m m ij
n
E ) should converge in numerical
1i < jm
sense for any such sequence En . Such functions f (calorime- Jm
tric, or C-observables) are exactly the ones that are stable with It turns out that 0 1 on any state if Nm is defined from
respect to measurement errors of calorimetric detectors. the condition J sym = sym
m (P ) 1 where P is a limit of uniformly
5. The role of C-continuity is best understood with the help sym = -1
of analogy with the familiar length measurements. Length is distributed states PN with N so that E N
i and
habitually represented as a real number--an idealization that, N d$ . Then N = , N = / , N = ,
i 4 z i
p 2 2 3 27 4 4 36
one tends to forget, is highly non-trivial from a historical per-
spective. In particular, the familiar continuity of real numbers N = =
5 9375 / 32 , N6 455625 / 128 .23
is useful only inasmuch as it corresponds to the stability of, Some special values of Jm(P ) are as follows. For the state
say, volume computations with respect to data errors of the P sym
3 consisting of 3 symmetrically arranged particles,
length measurements involved. The elusive reality of calorime- J (P sym ) = / . . For a symmetric state of 4 particles
tric measurements is that whereas rulers measure length as a 3 3 27 32 0 84
real number, calorimetric detectors measure energy flow as an (tetrahedron) J = =
3 ( 4
P sym) 1 , J4 ( 4
P sym) 64 / 81 . For a sym-

additive function on subsets, and the C-continuity plays exactly metric state of six particles (octahedron),
the same role for data errors of calorimeters as the usual conti- J sym sym
= =
3 (P ) J
6 4 ( 6
P ) 1 .
nuity of real numbers does for rulers. 1
6. A large class of C-observables is immediately found as
follows.17 Consider the direct product of m identical EFs
E( $p) . Then the standard theorems14 imply C-continuity of the

observables of the following form: ycut
F (E) = dp$ K dp$ E( p$ )KE( p$ ) f ( p$ ,K, p$ ) , (2)
m z 1 z m 1 m m 1 m
0
where fm is any continuous symmetric function. A function on m =1 2 3 4 5 6
EFs induces a function on multiparticle states: using (1) and
(2) one obtains: Fig. 1. The crosses are the values of the jet discriminators Jm for a typical
final state. When looked at sideways, the fat lines represent the "number of
F {
c E , p$ }h = E K E f ( p$ ,K, p$ )
. (3)
m i i i i
K i i m i i
1 m 1 n 1 m jets" as a function of ycut .
This is automatically fragmentation invariant. If fm satisfy Fig. 1 shows a typical picture of values of jet discriminators.
minimal requirements of regularity (e.g. existence of first de- The usual jet counting amounts to replacing the continuous Jm
rivatives), then Fm are IR finite18. Such C-observables are in- with 0 or 1 (the blobs). This can be achieved e.g. by introduc-
terpreted as average values of operators that are m-local in ing a cutoff ycut as shown.24 The non-zero tail at large m is
momentum space, which offers a possibility of their systematic due to hadronization (including Sudakov effects). The instabi-
theoretical study.19 Examples of C-observables are the well- lity with respect to such effects as well as to data errors (shifts
known thrust, spherocity, etc. (see Ref. 4 for a complete list).20 of the crosses) is clearly seen.25
Algebraic combinations of C-observables are again C- Note that J
m 0 for m larger than the number of particles
observables. But taking e.g. infinite sums of such functions (or detector cells). For decreasing width of jets and for
( m + ) requires care: one can arrive at observables that are m > M (a typical number of jets in the event), Jm are in-
IR safe in each order of perturbation theory, continuous in the
ordinary sense as functions of particles' energies and momenta creasingly suppressed by powers of 2 and of the energy frac-
for any fixed number of particles, but not C-continuous.21 A tions of soft particles. This ensures a monotonic decrease of the
complete understanding of this subtlety in a general QFT con- values of J >
m for m M for typical events. Numerical experi-
text is lacking. Anyhow, C-continuity limits available options, ments show that the decrease of Jm is a universal feature even
and if one wishes to deal with correlator-type observables then for m M .23,26
the above Fm remain the only choice. Fragmentation causes the values of Jm to increase as com-
7. MEASURING THE "NUMBER OF JETS". Imagine a step func- pared with the parton state. However, the C-continuity of Jm
tion equal to 0 on states with less than m jets, and to 1 else- ensures that the closer (in the calorimetric sense) the final had-
where. A sequence of such functions (m = 1, 2, ...) would do the ron state to the parton state, the less the difference in values of
job of jet counting just fine. But we wish to deal with C-obser- C-observables, and the less the upward shift of J
vables. So, consider a C-observable (3) that is exactly 0 on any m .
8. For the case of hadrons in the initial state one should
state with less than m particles. Then f p p =
m ( $ , $ ,K) 0 , so that
modify J
f p p m to suppress contributions from the hadron beams.
m ( $1, $2 ,K) should contain a nullifying factor 12 , e.g.: For pp , say, it is sufficient to introduce into j
m the factor
1 1
= - cos = - -
p p p P p P
12 1 12 ( 1 2)( 1 ) ( 2 ) , (4) 1 2
- cos i per each particle where i is the angle between the
particle's direction and the beam axis.


F.V.Tkachov Phys. Rev. Lett. 73 (1994) 2405; Erratum 74 (1995) 2618 3



9. Now, fix a multiparticle state P and consider any jet
counting algorithm A that produces an integer number of jets [4] R. Barlow, Rep. Prog. Phys. 36, 1067 (1993).
N y [5] For instance, one needs to identify jet axes to study the QCD coher-
A ( cut ; P ) for each ycut , which is non-decreasing as ence (Yu. L. Dokshitzer et al., Rev. Mod. Phys. 60, 373 (1988)), but
y
cut 0 . Then from Fig. 1 one sees that one could, in theory, the notion of jet's axis is ambiguous beyond the leading order of per-
restore a sequence of jet discriminators J A turbation theory.
m ( P ) similar to [6] Cf. F. V. Tkachov, in Ref. 7.
J A
m (P ) . Thus, the information content of Jm ( P ) and [7] Cf. Proc. New Techniques for Calculating Higher Order QCD Cor-
N y rections (ETH, Zrich, 16-18 December 1992), ed. Z. Kunszt (ETH,
A ( cut ; P ) is essentially equivalent. But it is hardly possible Zrich, 1992).
to find meaningful expressions for J A
m ( P ) for the popular al- [8] For a review see S. Bethke et al., Nucl. Phys. B370, 310 (1992) and
gorithms. Our J S. Catani, in: Proc. 17th INFN Eloisatron Project Workshop, Erice,
m (P ) are singled out by the transparency of 1991, ed. L. Cifarelli and Yu. L. Dokshitzer (Plenum Press, New
analytical structure. York).
10. So, studying the average values of jet discriminators [9] T. Sjstrand, Comp. Phys. Commun. 28, 229 (1983); see also Ref. 8.
Jm (qualitatively interpreted as fractions of events with no [10] Cf. W. T. Giele and E. W. N. Glover, Phys. Rev. D46, 198 (1991).
[11] A more stable variant of recombination was described in S. Youssef,
less than m jets) instead of the usual n -jet fractions may have Comp. Phys. Commun. 45, 423 (1987).
an advantage of reducing, in perspective, both theoretical and [12] Cross sections for adjacent numbers of jets differ by O( ) , so 1% of
S
experimental uncertainties: 3-jet events identified as having 4 jets (due to data errors or incom-
To compute Jm from data, one would treat each calorime- plete knowledge of hadronization) means an O(10%) error for the 4-
ter cell as a particle (the correctness of this is ensured by C- jet cross section.
[13] Multijet channels can be used e.g. for top search (F. A. Berends et al.,
continuity). Computations can be optimized due to the regular Nucl. Phys. B357, 32 (1991)). On the other hand, the study of multi-
structure of Jm in several ways: (i) One can do the summa- jet cross sections by UA2 (K. Jacobs, in "Joint Int. Lepton-Photon
tions  la Monte-Carlo with probabilities equal to energy frac- Symp. on HEP" (1991, Geneva) World Scientific: Singapore, 1992)
tions. (ii) A preclustering can be used due to C-continuity to concluded that the agreement of the data with theory they found for 4
6 jets "can be considered as largely accidental".
reduce the number of particles to, say, 30 when computa- [14] For a thorough treatment see L. Schwartz, Analyse Mathmatique,
tions are easily manageable; since the exact expression is vol. 1 (Hermann, Paris, 1967).
known, the approximation errors are fully under control here. [15] The word "measure" is used here in two different meanings, physical
(iii) The computation of (5) can be parallelized. and mathematical, not to be confused.
On the theoretical side, studying effects of hadronization [16] Note that all possible EFs form an infinitely-dimensional space, and
would reduce to studying logarithmic and power corrections to there are many radically non-equivalent ways to define convergence
(i.e. topology) in such spaces (see any textbook on functional analy-
Jm . Resummation of logarithms is done via the standard sis). To appreciate the subtlety of the problem recall the large-scale
renormalization group. The analytical calculations of the cor- study of G. C. Fox and S. Wolfram, Nucl. Phys. B149, 413 (1979),
responding diagrams are easier due to the simple analytical who adopted an incorrect idealization of EFs as functions on the unit
sphere with L
form of the weights in the phase space integrals (cf. (4)). Also, 2 topology (familiar from quantum mechanics) and the
research was lead astray towards studying spherical harmonics etc.
a prospect opens for a study of power corrections27. Recall that Our C-convergence is the well-known weak convergence of linear
the power corrections for + -
tot (e e hadrons) J2 are functionals. It cannot be described in terms of a single-valued distance
given by expressions involving vacuum condensates28 that are function or norm (cf. Ref. 14). This may be psychologically uncom-
directly related to soft singularities; the structure of power cor- fortable but such is the nature of calorimetric measurements.
rections can be obtained within perturbation theory29 while the [17] Cf. the cumbersome constructions of Fox and Wolfram, in Ref. 16.
[18] G. Sterman, Phys. Rev. D19, 3135 (1979).
values of condensates are estimated via the lattice QCD. A [19] Cf. F. R. Ore and G. Sterman, Nucl. Phys. B165, 93 (1980).
similar approach should be feasible for the jet discriminators.30 [20] Note that the energy correlations of C. C. Basham et al., Phys. Rev.
11. The importance of the problem of jet definition was im- Lett. 41, 1585 (1978) are a special case of Fm --but with a discon-
pressed upon me by S. D. Ellis. A crucial encouragement came tinuous fm . Since hadronization is not inverted here, the lack of C-
from A. V. Radyushkin. I thank Z. Kunszt and ETH for the continuity is numerically less important than with recombination algo-
hospitality during the Workshop on New Techniques for Cal- rithms.
culating Higher Order QCD Corrections (ETH, Zrich, De- [21] F. V. Tkachov, in "Joint International Workshop on High Energy
cember 1992)--its atmosphere catalyzed the present work. Physics" (September 1993, Zvenigorod, Russia), ed.
Yu. Bashmakov, S. Catani, R. K. Ellis, I. F. Ginzburg, B. B. Levtchenko (INP MSU: Moscow, 1994), p. 80.
[22] For e+e- this is the total 4-momentum. For ep one may wish to
B. L. Ioffe, A. Klatchko, Z. Kunszt, D. V. Shirkov, choose a different frame--as described e.g. by B. Webber, J. Phys.
T. Sjstrand, I. K. Sobolev, B. Straub, S. Youssef and the three G19, 1567 (1993).
referees supplied bibliography and/or necessary criticisms. I [23] I thank B. B. Levtchenko for numerical checks of this.
thank the participants of several workshops and seminars for [24] Such a jet counting procedure was called Moscow sieve in
lively discussions, and J. C. Collins and H. Grotch for the hos- F.V. Tkachov, technical report, INR-F5T/93-01 (unpublished).
pitality at the Penn State University where this work was com- [25] The figure suggests that the importance of Sudakov effects in the con-
ventional recombination algorithms is an artefact due to the instability
pleted. It was supported in parts by the U.S. Department of En- of the latter.
ergy (grant DE-FG02-90ER-40577) and by the International [26] S. Catani et al., Nucl. Phys. B406, 187 (1993) discuss relationship
Science Foundation (grant MP9000). between jet clustering algorithms and event shape measures, including
the monotonicity.
* On leave of absence from the Institute for Nuclear Research of Rus- [27] whose importance is discussed by B. W. Webber, in Ref. 7.
sian Academy of Sciences, Moscow 117312, Russia. [28] M. A. Shifman et al, Nucl. Phys. B147, 448 (1979).
[1] For a review see S. D. Ellis, "Lectures on perturbative QCD, jets and [29] F. V. Tkachov, Phys. Lett. B125, 85 (1983).
the Standard Model: collider phenomenology" at the 1987 Theoretical [30] The required mathematics is reviewed in Ref. 6.
Advanced Study Institute (1987, July, St. John's College, Santa Fe,
NM), available as technical report NSF-ITP-88-55.
[2] G. Hanson et al., Phys. Rev. Lett. 35, 1609 (1975).
[3] G. Sterman and S. Weinberg, Phys. Rev. Lett. 39, 1436 (1977).



