1.1. Phase Space Volume

International Journal of Mo dern Physics A

f

cWorld Scientic Publishing Company

JETS AND KINEMATICS IN HADRONIC COLLISIONS

ANDREW R BADEN

Physics Department University of Maryland

Col lege Park MD USA

Received September

Revised Decemb er

Analysis of collidingb eam hadron exp eriments p p X and p p X often dep end

up on observation of jets a highly collimated spray of particles suchas Keand

s forming clusters of energy dep osition in calorimeters In this pap er we outline howto

dene jets from such clusters and discuss the meaning of jet quantities such as transverse

energy and mass Data from the D exp eriment are used to illustrate these concepts In

addition we review the motivations for using certain optimal co ordinates for describing

energymomentum vectors and deriveinteresting relationships among the kinematic

variables

Kinematics of Hadron Collider Events

To motivate some of the kinematic quantities traditionally used to describ e p p

p

exp eriments one must rst recognize that the centerofmass energy s of the

scattering in these exp eriments centers is not xed This is a direct consequence of

the proton substructure where the scattering constituents are quarks and gluons

that reside inside the nucleons with each parton carrying a fraction x of the total

nucleon energy Scattering of partons of dierent energy results in a centerof

momentum that do es not necessarily coincide with the lab oratory frame A lackof

knowledge of the centerofmass energy guides the choice of kinematic variables

Phase SpaceVolume

We start with the denition of a relativistically invariant phase space volume d

for an ob ject of momentum p and energy E For a xed particle mass d can b e

written as see

dp dp dp d p

x y z

d

E E

Dening z as the pp collision axis we see that of the momentum co ordinates

p p p and E only the rst two are invariant with resp ect to a Lorentz transfor

x y z

mation from the lab frame along the z direction by virtue of their direction b eing

normal to that of the b o ost In addition we see that even though the co ordinates p

and are not invariant with resp ect to b o osts along zinvariance will b e preserved

in the pro duct p cos sin whichisp

x

Achoice of momentum co ordinates which more explicitly reect the Lorentz

invariance of the phase space elementisp ym where p is the momentum

T T

transverse to z direction is the azimuthal angle ab out z m is the mass and y is

the rapidity along the z direction dened as

Jets and Kinematics in Hadronic Col lisions

E p cos

z

y ln ln

E p cos

z

or equivalently

cos tanh y

where pE and is as ab ove the p olar pro duction angle relativetothez

axis For a b o ost along thez direction with some a transformation from

b

the lab frame to the new frame will change the rapidity as follows

y y y

b

where

y ln

b b b

Because cos for b o osts along z it follows that

tanhy

b b

The advantage of using the ab ove set of co ordinates is clear the eect of a b o ost

along the z axis changes only y and that by only an additive constant

To calculate d in these co ordinates we rst dierentiate Eq

y y E

dy dp

z

p E p

z z

p p E

z z

dp

z

E p E p E

z z

which simplies to

dp

z

dy

E

The comp onents dp and dp are related to the elements dp and d via a trans

x y T

formation from cartesian to p olar co ordinates in dimensions

d dp dp p dp d dp

x y T T

T

We can now write the volume element d as

dp dy d d

T

Thus particles pro duced in pro cesses that have a matrix elementthatvaries slowly

with rapidityy should b e distributed uniformly in y This is the origin of the

rapidity plateau for singleparticle pro duction in hadronic collisions

Our chosen set of kinematic co ordinates is well suited to the analysis of physics

issues and is used extensively to describ e the results of measurements in high energy

physics In the following subsections wedevelop formulae and relationships among

p y andm and other quantities of interest

T

Transverse Energy E and Momentum p

T T

One can dene the transverse energy E of a particle as its energy in the rest

T

frame where its p Because this quantityisinvariant to boosts along zwe can

z

write in general

E p p m p m E p

T x y T z

From Eq we can solve for p yielding

z

p E tanh y

z

Combining the ab ovetwo equations gives

E E E tanh y

T

E tanh y

E cosh y

or equiv alently

E E cosh y

T

Note that the p is always dened in terms of the momentum p as

T

p p sin

T

We wish to stress that these two relativistically invariantquantities E and p

T T

are not dened in the same way that is E E sin More on this b elow

T

Combining Eq and gives

p E sinh y

z T

Invariant Mass

The invariant mass of two particles is dened as

M p p p p

E E p p m m

Using the p y and mvariables and applying equations from the pro ceeding

T

sections we calculate

E E E cosh y E cosh y

T T

and

p p p p p p p p

x x y y z z

p cos p cos p sin p sin p p

T T T T z z

p p cos p p

T T z z

p p cos E sinh y E sinh y

T T T T

E E cos sinh y sinh y

T T T T

where p E sin cosh y and

T T T

Combining the ab ove results gives

M m m E E cosh y cos

t t t t

where y y y and cosh y cosh y cosh y sinh y sinh y We will discuss

the consequences of Eq in the limit in a subsequent section

PseudoRapidity

In order to determine the rapidity of a particle one needs b oth its cos and velo city

Since the velo city requires measuring b oth E and the mass m this precludes

using a single detector comp onent to measure rapidityHowever in the limit of

or m p Eq b ecomes cos tanhy which often pro vides an

T

excellentapproximation to use for the rapidityWe therefore dene the pseudo

rapidity as the rapidity of a particle with zero mass or y j

m

Using Eq for y and setting one obtains

cos

ln ln tan

cos

and the inverse equation which is often useful

cos tanh

Eliminating cos from Eqs and wehave the relationship

tanh y tanh

This last equation shows clearly that in the limit m or wehave y

The pseudorapidity dened to b e indep endentof is therefore another wayto

measure the lo cation of a particle equivalent to measuring its p olar angle In

addition we can use Eq to derive the transformation

d sin d d cosh

Since is always and the function tanhx is monotonic the pseudorapidity

j j of a particle of given p is larger than its true rapidity jy j Figure shows the

T

dierence b etween and y as a function of for protons for several choices of

constant p and Figure shows a similar plot for pions In each plot a dashed

T

line is drawn at constant j jjy j corresp onding to the size of the calorimeter

readout pad at D Rapidity and pseudorapidity are equal to within for pion

p of GeVc and proton p of GeVc for all This means that one

T T

can use in place of y in kinematic calculations involving pions and protons in this

range of p values

T

It is imp ortant to recall that for xed p the velo city is a function of pseudo

T

p

rapidity To see the explicit form we start with pE E p m use

Eq for p and Eq to relate to to get

T

p

T

p

m tanh E

T

which gives

p tanh

T

q

tanh y

m tanh E

T

Given the fact that j jjy jwewould exp ect that for xed p any single

T

particle sp ectrum that is indep endentofy d dy is constant would b ecome de

pleted in the central region when plotting as a function of Thistyp e of eect

would b e esp ecially large for particles with low p and signicant mass T

Fig j j jy j for protons at various p

T

Fig j j jy j for pions at various p T

d dy dy d

k

d dy d d

where k is a constant An analytic form for dy d can b e derived starting with Eq

substituting for using Eq dierentiating and rearranging terms to get

the result that

dy

d

with given by Eq This result although somewhat surprising can b e

understo o d in the following sense at large y and therefore large the function

tanh y converges to Hence for large y b oth the tanh y and tanh converge to

which means whichwould b e true for large p and xed p In this case

T

there is little if any functional dep endence of y on Atsmally we can use the

approximation tanh y y and tanh to get the relation y whichgives

dy d We therefore exp ect dy d to b e constant for large y or with

all of the dep endence coming at small

Figure shows a histogram of dN d for pions generated with constant

p GeVc and uniformly in dN dy The smo oth curve is the function

T

that is dy d for pions with xed p GeVc

T

Fig The histogram shows dy d for pions generated uniformly in rapidity y with xed p

T

GeVc The smo oth curve is the corresp onding function

Invariant Masses for

In the limit m y and We can then write the invariantmass

T

of two massless particles using Eq as

E E cosh cos M

T T

This expression is often quite useful as weshowbelow

Transverse Mass and Missing Transverse Energy

Noninteracting particles such as neutrinos can b e detected only via an apparent

imbalance of momentum in an event In p p or ppinteractions where most of the

longitudinal parallel to the b eam z momentum in the collision is lost down the

b eam pip e it is essentially imp ossible to measure momentum conservation along z

However given a detector with suciently large acceptance measuring momentum

balance in the plane transverse to the b eam direction is relatively straightforward

Thus that amount of transverse momentum required to satisfy momentum con

servation the missing transverse momentum can b e in large part attributed to

the presence of any unobserved noninteracting particles esp ecially when the total

momentum and energy in the detector is large since for small total detected en

ergy small uctuations in the measured momentum along anygiven direction can

give a small statistically insignicant missing transverse momentum Since the

neutrino is massless certainly on these scales this quantity missing transverse

momentum is virtually equivalenttothe missing transverse energy or E in the

T

event

For particles such as the W b oson which decayinto leptonneutrino pairs the

lack of a measurementofp for the neutrino precludes a direct measurementofthe

z

invariant mass of the W However one can calculate an in variant transverse mass of

these two particles denoted by M dened as the invariant particle mass using

T

only the momentum comp onents in the transverse plane p and p i From

xi yi

this denition the transverse mass is dened as

M p p p p

T T

T

T T

where p E p For two zeromass particles E p wehave

Ti Ti Ti Ti

Ti

E E p p M

T T T T

T

E E cos

T T

E E sin

T T

or equivalently

p

M sin E E

T T T

We can see that this formula is equivalent to Eq with the condition

that is the transverse mass and the invariant mass are equivalent for particles which

are pro duced at the same rapidity As such M M ignoring the

T

detector resolution and intrinsic width As a rule M will deviate from M

W T

as gets large that is as the particles decay more forwardbackward in the lab

frame Distributions in M will therefore have a tail at low M forwardbackward

T T

will p eak at M and will fall to zero ab ove M in the same manner as M

It is imp ortant to note that these formulae require that E b e dened using Eq

T

as E sin They are equivalent only in the limit mp Of course and not

T

Eq is true in general As will b e shown b elow in the small mass limit of

the dierence b etween E and p b ecomes insignicant

T T

An interesting relationship involving the invariant mass of particles is

M M M M m m m

which in the limit of m becomes i

M M M M

This formula is interesting to keep in mind when searching for b o dy decays say

A a b c If it is appropriate to ignore the three masses m m andm then

a b c

you do not havetoknowwhich of the three b o dies is a bor c sp ecically in order

to calculate the correct value for m you only havetoknow that these are the

A

b o dies of whichyou are interested

Lastly since the mass of the W particle is well known we can constrain the

invariantmassofthe e pair and solve for the longitudinal momentum of the

neutrino To do this we can use Eq

M E E cosh cos

T T

W

Rewriting this expression weget

M

W

cosh cos

E E

T T

Solving for gives

p

r r

ln

where r is the righthand side of Eq Because is the dierence in pseu

dorapiditybetween the electron and the neutrino there are two solutions to the

problem That is there is no way of resolving the ambiguity of whether the neu

trino is at a lower or higher rapidity relative to the electron as seen from the fact

that the hyp erb olic cosine cosh is even in Both solutions are p ossible at least

in principle

Jets

Quantum Chromo dynamics QCD is the currently accepted theory of the strong

interaction b etween particles partons that carry color These partons quarks q

and gluons g app ear in detectors as sprays or jets of particles In this section

we will discuss issues concerning the observation of jets and their origin in QCD

Denition of a Jet

There are various ways to dene jets For instance most of the LEPPEPand

PETRA e e exp eriments dene jets using information just from tracking cham

bers One of the less ambiguous ways to dene jets is motivated bywhatwe

understand to b e the physics of fragmentation namely that a jet is the result of

the hadronization of a parent parton A way to visualize hadronization is through

wed virtual gluon emission by the parentpartonq q g and g g g follo

by gluon splitting to q q pairs forming the color singlet hadrons that we measure

For high precision studies of parton QCD fragmentation in e e collisions this

mo del is to o naive However in hadronic collisions where the presence of the soft

underlying event part of a high Q collision complicates the analysis the naive

mo del is adequate for describing how a parton turns into a jet

To see how this simple fragmentation mo del can b e related to the concept of

a jet we consider the transverse momentum k of radiated gluons relativeto

T

the jet axis The distribution of k has b een found to b e exp onential in k with

T

p

and DORIS energies hk i MeVc at SPEAR s GeV increasing

T

to hk iGeV c at SppS energies We dene as the angle of the gluon with

T

resp ect to some reference in the plane p erp endicular to the momentum direction of

Fig Radiated gluons make an angle with resp ect to an arbitrary direction in the plane

p erp endicular to the jet axis

the parent parton see Figure Since the physics of gluon radiation is indep endent

of this angle we exp ect the probability distribution for will b e uniform in The

jet axis can b e dened by a cluster of calorimeter cells less often bytracks in the

tracking chamb er in space that minimize the total vector k of the jet remnants

T

This means requiring

X

k

Ti

We can use Eq to form the jet cluster into an equivalent particle with

momentum p E p that can b e used for example to calculate an invariant

mass for example W jet jet Let us sp ecialize to calorimetry and assume that

the jet will b e constructed as a cluster of cells or towers for pro jective geometry

calorimeters If one represents the energy E in each cell of a calorimeter i as a

i

particle with zero mass one can then dene E E n where n is the unit vector

i i i i

pointing from the interaction vertex to cell i The jet is therefore dened in the rest

frame of the detector as an ob ject consisting of a set of these zero mass particles

E one p er calorimeter cell Note that this representation of and therefore p

i i

a jet using zero mass particles do es not imply that the jet also has zero mass or

p E since the constituents the cells will eachhave an op ening angle b etween

them and this op ening angle will generate a jet mass To see this explicitly

consider Eq the invariant mass of two particles If the particles have zero mass

we drop the terms m and m set E p and E p leaving

M E E cos E E sin

where is the angle b etween the two particles We see clearly that if the

invariant mass is also zero

To form a jet from these calorimeter cells wemust apply Eq relativeto

the axis that denes the jet direction The transformation of the cells to this axis

involves an Euler rotation with degrees of freedom and The rotation

jet jet

matrix M dened by E M E is a function of the yet to b e determined Euler

i

angles and is given by

sin cos

jet jet

A

cos cos cos sin sin

jet jet jet jet jet

sin cos sin sin cos

jet jet jet jet jet

The transformation equations which relate the cell energies in the frame of the jet

primed to those in the frame of the detector unprimed are therefore

E E sin E cos

xi jet yi jet

xi

E sin

Ti i

E cos cos E cos sin E sin E

xi jet jet yi jet jet zi jet

yi

cos E cos sin E

jet Ti i jet zi

E E sin cos E sin sin E cos

xi jet jet yi jet jet zi jet

zi

sin E cos cos E

jet Ti i jet zi

where E E sin and As a check of Eq we can

Ti i i i i jet

imagine a jet with a single tower which when rotated to the jet axis should retrieve

E E and E E Thus setting remember that is the

jet

x y z

dierence in between the jet and the tower the ab ove equations yield E

x

E E sin E cos using tan E E and E E sin

z jet T jet jet T z T jet

y z

sin and E E cos Here wehave E cos E using E E

jet z jet z jet jet T

assumed that suchahyp othetical jet would b e massless since it is contained in a

single tower allowing E p in the ab ove This is discussed further b elow

jet jet

The momentum vectors of eachtower can now b e rotated into the frame of

the jet summed over the transverse comp onents E k and E k and

xi yi

separately set to zero as required by Eq

X X X

E sin E cos E

jet xi jet yi

xi

i i i

and

X X

E cos cos E

xi jet jet

yi

i i

X X

E sin E cos sin

yi jet zi jet jet

i i

We therefore havetwo equations and and two unknowns and

jet jet

Solving Eq yields the following relation

P P

E sin E

Ti i yi

P P

tan

jet

E E cos

xi Ti i

Solving Eq gives

P P

cos E sin E

jet xi jet yi

i i

P

tan

jet

E

zi

i

Equations and describ e the co ordinates of the jet cluster in a waywhich

is consistent with the dening Eq

The jet is a true physical ob ject which mighthave originated through the show

ering of some virtual parton and it must therefore haveavector E p that

jet jet

corresp onds to an ob ject of some nite mass We can dene the angles of our jet in

the lab oratoryasfollows

p

yj et

tan

jet

p

xj et

and

p

tj et

tan

jet

p

zj et

or equivalently using the relation that sinh tan

p

zj et

sinh

jet

p

tj et

Equating the numerators and denominators of Eqs and we obtain the

result

tow er s

X

p E

xj et xi

i

tow er s

X

E sin cos

i i i

i

tow er s

X

E cos

Ti i

i

and

tow er s

X

p E

yj et yi

i

tow er s

X

E sin sin

i i i

i

tow er s

X

E sin

Ti i

i

By comparing the numerators and denominators of Eqs and weget

towers

X

p E

zj et zi

i

towers

X

E cos

i i

i

towers

X

E tanh

i i

i

towers

X

E sinh

Ti i

i

where wehave again used the relationship sinh tan andnotshown the details

of the algebra Wehave also used the fact that for the towers E E sin

Ti i i

rememb ering that in our approximation each calorimeter tower b e represented as a

particle of zero mass

It should b e recognized that the co ordinate was derived from Eq

jet

and the co ordinate from Eq using the relationship log tan

jet

Consequently the true do es not corresp ond to the energy weighted result

jet

P P

E E

Ti i Ti

The energy of the jet is given by an energy sum over all towers in the cluster

tow er s

X

E E

i jet

i

Wehave therefore established a selfconsistent prescription for turning a cluster

of energy towers into a physical jet which is a particle with momentum in the

detector frame given by

X X X X

p E p E E n

i i i i i

jet

and with directional co ordinates or and derived from the ab ove

jet jet jet

momentum comp onents or as calculated using Eq through

It is worth investigating expressions for the jet vector that involve not just

sums o ver towers at sp ecic angles and but rather sums that involve angles

i i

relative to the jet axis and

i jet i i jet i

The expression for p is straightforward Weusethenumerator of Eq

Tjet

X X

p cos E sin E

Tjet jet xi jet yi

i i

and substitute in E E cos and E E sin Thisgives

xi Ti i yi Ti i

X X

sin E cos sin E p cos

i Ti i jet Ti Tjet jet

i i

X

E cos

Ti i i

The expression for E is not as straightforward First we form E E p

Tjet

z

T

and use the relation cosh x sinh x to get

E E p

Tjet jet zj et

E cosh sinh p cosh sinh

jet jet jet zj et jet jet

E cosh p sinh E sinh p cosh

jet jet zj et jet jet jet zj et jet

If we add and subtract the quantityE p sinh cosh and then use Eqs

jet zj et jet jet

and in the ab ove equation weget

E E cosh p sinh E sinh p cosh

jet jet zj et jet jet zj et jet

Tjet

X X X X

E cosh E sinh E sinh E cosh

i jet zi jet i jet zi jet

X X

E cosh cosh E sinh sinh

Ti i jet Ti i jet

X X

E cosh sinh E sinh cosh

Ti i jet Ti i jet

X X

E cosh E sinh

Ti i Ti i

This equation is not altogether transparent However consider the symmetry of

the sinh and cosh functions the former is o dd the latter is even If the jet is

symmetric in energy and physical extent ab out the jet axis then the sum over

E sinh will vanish leaving

Ti i

X

E E cosh

Tjet Ti i

Eqs and will prove useful later in our exp osition

Jet or Clustering Algorithm

The nal missing ingredient is an algorithm for determining which calorimeter cells

to include in a cluster that is consistent with Eq One suchscheme usually

referred to as a xedcone algorithm used byUA CDF and D works

in the following way

All towers ab ove a mo derate energy threshold serve as seed towers for initial

protoclusters

All immediately neighb oring towers within some cone cuto R in space

a common range for the cone cuto R is relative to the center of

p

are added to form a new cluster The the protocluster R

centroids or st moments of the cluster in and are recalculated These

towers mayormay note b e sub ject to a small energy threshold

If the jet centroid in changes by more than some small preset value then

all towers within the cone R are recombined to form a new cluster Again

the centroids are calculated in and This pro cedure is iterated until the

centroid of the cluster remains stable

After all clusters are found some maycontain overlapping same calorimeter

towers If so such clusters are merged dep ending on the fraction of transverse

energy present in the shared towers The smaller E cluster is then subsumed T

A threshold parameter controls the merging Forthedatatobeshown in this

rep ort the merging parameter was set to of the energy shared by

anytwo clusters will cause merging

Another algorithm called the nearestneighb or algorithm was the primary clus

tering algorithm used by UA and is describ ed elsewhere

The Shape of a Jet

Equation can b e used to calculate the comp onent of the tower energy or

momentum that is transverse to the jet direction k E E For those

xi yi

Ti

towers that are close to the jet axis the smallangle limit of Eq gives

E E

Ti i

xi

E sin E cos E

jet zi jet Ti

yi

sin E cos cos E sin

jet i i jet i i

E sin

i i

E

i i

E

Ti i

where wehave used sin from Eq and the relation E E sin

Ti i i

Squaring and adding the comp onents gives

k E

Ti Ti i i

or equivalently

k E R

Ti Ti i

where R is the distance b etween the jet and cell i in space dened by

i

R

i i i

Thus the momentum of each particle in the plane is given by the pro duct of

the E of the particle in the lab frame and the distance in space from the jet

T

axis Since the k are exp ected to b e symmetrically distributed ab out the jet axis

Ti

ve circular shap es in space Of course this Figure this suggests that jets ha

small angle limit is not a go o d approximation for particles cells that are far o

the jet axis

It is b est to dene the shap e of a jet using an energy weighting pro cedure This

reduces the dep endence of the shap e on low energy towers that are b oth far from

the jet axis and have a negligible contribution to the total jet energy The shap e

of a jet is often characterized byanE weighted second momentvariance of the

Ti

distance in the transverse plane Since the rst momentvanishes

X X

k E R

Ti Ti i

wehave

P P

E E R

Ti Ti

i i i

P P

R

E E

Ti Ti

which is equivalentto

R

when we dene and as

P P

E E

Ti Ti

i i

P P

E E

Ti Ti

Referring to data for some sp ecic R we rst dene

q

p

R

using events from D describ ed in App endix Sample This data sample was

selected to minimize jet merging to a pro cess that has origin in higherorder

QCD eects that are distinct from those of jet fragmentation

Figure shows the distribution of the second moment for three jet cone

R

cutos of R and The p oints are D data The eect of the cone

cuto is clear the larger the cone the larger the nd moment in b oth mean and

width The main source of the broadening is simply due to the presence of additional

towers for the larger cone cutos Figure also shows the results of Gaussian ts

to the distributions The ts reveal that for unmerged jets the distribution of jet

widths is well describ ed by a Gaussian with a central value that is a function of

the cone cuto This is not to say that the energy ow around the jet axis has a

Gaussian distribution rather this is only describing the uctuations in the jet

R

RMS around the average which is itself a function of the jet cone parameter

R

Table shows the results for the ts and Figure shows that the central value of

Table Results of a Gaussian t to distributions of using the data from Sample

R

R RMS

R R

the distribution in mean of the Gaussian scales linearly with the cone cuto

R

The small angle approximation used ab ove is clearly excellent Figure shows

that even for jets with R ab out of the time This shows that

R

the ab ove approximationisvalid for all but p erhaps of the R jets with

largest RMS widths

Jet Correlation Matrix

Detector eects gluon radiation and the underlying event all contribute to dis

torting the shap e of jets Figure shows a sketchofatypical elliptical real jet

with dened as the smaller of the angles b etween the semima jor axis and either

the or axis The angle can b e calculated from the variance in and as

dened in Eq and from the corresp onding correlation term dened as

P

E

Ti i i

P

E

Ti

The shap e of the jet can b e characterized by the correlation term and the

diagonal terms and in the shap e matrix

Up on diagonalization the two eigenvalues corresp onding to the semima jor

and semiminor axes are

Fig Jet probability distribution in for single unmerged jets

R

Fig Results of a Gaussian t to distributions in as a function of the cone cuto RThe

R

straight line is from a linear t to the p oints yielding R R

η

α φ

Fig Jet ellipse Semima jor and minor axes represent the square ro ot of the second moment

in and

s

It is clear that for a circular jet and

Wenow return to Eq which dened the quantity the RM S width of

R

the jet in space This quantity is of course the square ro ot of the sum of the two

diagonal terms of the shap e matrix and not the ro ot of the sum of the squares of

these two terms The odiagonal term do es not aect

R

The E of a Jet

T

A common way for exp erimenters to characterize the hard scattering energy scale

Q for QCD collisions which pro duce jets is to use the transverse energy E of

T

a pro duced jet The measure of E which is most often used to compare data to

T

QCD theory is

tow er s

X

E E

T Ti

i

It has the advantage that one can hop e to establish a corresp ondence b etween the

energy observed in eachtower exp erimen t and the particles radiated by a jet a

hadronized parton There are however two other ways to dene E one of which

T

is more amenable for treating a jet as a collection of physical ob jects particles

and p ossibly more appropriate for calculating invariant masses involving jets The

rst is

E E sin

T jet jet

where sin is related to through Eq and E is calculated using Eq

jet jet jet

The second way to dene E uses Eqs and As long as there is

T

consistency in comparing theory to exp eriment it is relatively arbitrary howone

denes E given that all internal variables are calculated consistently However

T

b ecause of the dierences b etween p and E the denition in Eq should not

T T

b e used in calculating the invariantmassgiven in Eq For such cases only

the E dened in Eq is appropriate

T

To see this clearlywe use the relation E p m and write

E sin p m sin p m sin

T

We see that this denition gives a systematical ly lower value for the transverse

energy E than the relativistically correct denition from Eq The dierence

T

of course is insignicant in the high energy limit jet velo city and for jets

that are pro duced at to the b eam axis socalled central jets

Wecanquantify the dierence in the two denitions of E through the quantity

T

q

p m sin

T E sin

p

E

T p m

T

which measures the fractional error in using Eq Expanding Eq for

small mass and keeping only the lowest p owers of m p gives

T

m tanh

p T

This shows the explicit dep endence of the dierence on p for arbitrary values

T

of For small tanh and rapidly approac hes zero Coupled with the

fact that mE for all but the smallestE jets we conclude that typicallyEq

T T

and Eq agree at the few p ercentlevel for

Figure shows the distribution in for the highest E jet in jet events from

T

our Sample App endix Figure shows the integrated probability as a function

of a lower cuto on the p ercen tage of jets that would b e mismeasured if w ere

greater than the value given on the abscissa The distributions clearly broaden with

jet cone size Because is prop ortional to m the broadening can b e attributed

to the broadening of the mass of a jet due to the increase in the numberofjet

towers with the cone size including towers that might not b elong in the jet

Such broadening can also come ab out from the failure to merge twojetsinto one

where one of the two jets mighthave an energy b elowanE threshold Sucha

T

threshold is usually employed by the exp eriments to eliminate the numb er of jets

found at very low energy

Fig Distribution in as dened using Eq for jets of Sample required to have j j

The E dened in Eq also underestimates the true E of the jet as dened

T T

using Eq This can b e seen as follows Supp ose there is a cluster with just

towers then Eq would yield

E E p p E p

z z

z

E E E cos E cos

E E E E cos cos

T T

while equation would yield

R R

dP where P is the distribution in Figure Jets were dP Fig I

required to have j j

X

E E E

Ti t t

i

E sin E sin

E E sin sin E E

T T

If we form the dierence of the two expressions weget

X

E p E E E sin

Ti

z

i

where We see that this dierence is always positive which means that

using Eq underestimates the true E of a jet

T

Again we form the quantity

P

E

Ti

i

E

T

whichshows the fractional error in using Eq for the transverse energyas

opp osed to Eqs and for the true E Figure shows the distribution

T

in as so dened As can b e seen in the gure the t wo dierent denitions of

transverse energy give similar results Dierences are not large and in fact Eq

gives a smaller error than one obtains when using Eq

The Mass of a Jet

Fig Same as Figure E dened using Eq

T

The mass of a jet can b e calculated from its transverse momentum and energy using

Eqs and

p E M

jet jet jet

E p

Tjet Tjet

E p E p

Tjet Tjet Tjet Tjet

Note that in the ab ove equation we formed the mass by using either E p or

E p One should not confuse the mass of the jet as dened by E p with

T T T T

what wehave previously describ ed as the transverse mass In fact the transverse

mass is only dened when dealing with or more vectors A jet do es not have

a transverse mass

In the limit when the jet energy gets large the jet narrows and

i

We expand Eq for small and Eq for small yielding

i i i

X

i

E p

Ti Tjet

and

X

i

E E

Tjet Ti

Calculating the dierence and sum b etween E and p wehave

Tjet Tjet

X

E p E

Tjet Tjet Ti

i i

X

E p E

Tjet Tjet Ti

i i

X

E

Ti

where wehave ignored the terms in and in the second equation Inserting

i i

these relations into Eq gives

X X

M E E

Ti Ti

jet i i

Nowwe pro ceed to the small angle thin jet limit for largeE jets As in the

T

P

sections ab ove we can use the approximation E E and combine with

Tjet Ti

Eq to obtain the relation for the invariant mass of the jet as

M E

jet R Tjet

This equation conrms what we knowintuitively ab out the invariant mass of a

multiparticle system namely that the mass of the system is generated by the

angles b etween the approximately zero mass particles

ws the normalized distribution in mass M for jets that were Figure sho

jet

reconstructed with cone cutos of and The distributions broaden as

the cuto increases due to inclusion of more towers further away from the jet center

Fig Mass distribution for jets of dierent cone size

In Figure we plot the ratio E M onaneventbyevent basis These

R Tjet jet

distributions show that the ratio is well within the few p ercent level of unity for all

jet cones note the vertical log scale used

Jet Merging

Fig The distribution in the ratio E M for dierent cone sizes

jet

R Tjet

The merging of two jets is usually reected in the value of and of course in

R

the size of the elements of the shap e matrix To study merging wehave used data

from D paying particular attention to whether there is evidence of any merging or

splitting of jets as controlled by the jet algorithm parameters see section These

data constitute our samples as describ ed in the app endices

Figure shows the normalized distributions in for merged and not merged

R

jets using our Sample for R and cone sizes We see a marked in

crease in the width of the merged jets for all cone cutos This sample is somewhat

biased by the requirement that after merging there must b e and only jets for

all three dening cone sizes

Figure shows the normalized distributions in using jets with cone cutos

R

of and for events from Sample App endix B This sample was chosen

to maximize the probability of collecting merged jets unlike Sample whichwas

chosen to maximize the probability of collecting unmerged jets The crosses and

the dots are resp ectively sp ectra for the merged and unmerged jets resp ectively

R

of Figure It should b e understo o d that each jet with R in Sample

is required to b e reduced to jets of R Figureshows that

R

correlates well on average with whether or not a jet is the result of a merging

We can therefore use the distributions from Figure to construct a likeliho o d for

a jet b eing from a merged or unmerged sample Figure shows the ratio of the

probability distributions for nonmerged and merged jets versus for cones of

R

and The p oints indicate the values of for relativelikeliho o ds of equal

R

probability of b eing from either distribution these are listed in Table

Finallyweinvestigate the shap e of jets via correlations in with resp ect to

merging If jets were circular in space wewould exp ect that the quantity

p p

would b e Gaussiandistributed around If jets result from merging

then we exp ect the shap e of the jet in space to b ecome elongated forming an

ellipse and the quantity would deviate from We use the normalized quantity

Fig Normalized distributions in for jets that were merged b ottom and those that have

R

not b een merged top for dierent cone sizes

Fig Normalized distributionsof for single jets that are asso ciated with jets found by

R

way of a smaller cone cuto see text

Fig Relative likeliho o d for a jet b eing nonmerged to a jet b eing merged as a function of

R

for three jet cones Relativelikeliho o ds of are indicated by the p oints on the curves

Table Value for corresp onding to a a relative likeliho o d of unity for jets to b e from non

R

merged or merged distributions

R

which to rst order should b e indep endent of the cone size for symmetric jets

This is indeed observed in Figure where weshow for jets from Sample

with cones and Figure shows the same distribution from Sample

Fig The distribution in using unmerged jets with cone cutos of and

from Sample

As demonstrated ab ove these jets are broader than pure unmerged jets however

Figure also shows that the broadening of can b e asso ciated with a broadening

R

in either or

Summary

The ab ove sections are intended to give a p edagogic description of jets towards an

understanding which is self consistent and illuminating For the particle physicist

the next step is to learn more ab out the physics of jets ie characteristics of jet

pro duction which in turn will allow understanding of the fundamental pro cesses

underlying the physics of interest

App endix A Events Used in Sample

A sample of events was used with the following requirements

i Pathologies Main Ring splash cosmic rays etcwere eliminated

ii The p osition of the collision vertex was required b e within cm of the

longitudinal center of the D detector

iii Eacheventhadtohave and only reconstructed jets for cones of

and with none of the jets formed from merging of twoormorejetsoflower

using merged jets with cone cutos of and Fig The distribution in

from Sample

transverse energy by merging we mean that there was no explicit merging

by the jet cone algorithm as describ ed ab ove

iv All electron and photon candidates are rejected by requiring a minimum

for cluster and dep ending on the cone size The cutos used were

and for cones of and resp ectively

These requirements were chosen to maximize the probability of selecting events with

only jets at tree level Nevertheless we cannot exclude the presence of events with

or more jets where jets were merged two at a time a priori due to their extreme

closeness in space Suchcontamination should b e small In addition there are

probably small biases due to eg trigger thresholds since the D jet trigger

starts with a single trigger tower threshold thinner jets are more likely than fat

jets to pass the trigger requirement and primary vertex cuts item ii ab ove is not a

very restrictive cut However the results presented here are intended to b e purely

qualitative in nature

App endix B Events selected for Sample

These events were chosen to have a sample of merged jets Events were required to

satisfy the same criteria and used to dene Sample but only events that had

jets reconstructed with a cone size of and jets with a cone size of or

were analyzed guaranteeing the presence of twowell separated clusters of energy

within a jet Ab out of all events were classied in this manner

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As of this submission the most precise published value of the W mass is

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most recent LEP data the W mass changes to GeV c These re

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httpsbhepphysicssunysbedu hadronspro ceedin gshtml

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