18-Jan-2006 UCLA 1

Jets, Kinematics, and Other Variables

A Tutorial

Drew BadenUniversity of Maryland

World Scientific Int.J.Mod.Phys.A13:1817-1845,1998

18-Jan-2006 UCLA 2

Coordinates

x

y

zProton beam direction

Proton or anti-proton beam direction

Detector

r

Protons Anti-Protons

E

Transverse E ET

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Nucleon-nucleon Scattering

b >> 2 rp

• Forward-forward scattering, no disassociation

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Single-diffractive scattering

• One of the 2 nucleons disassociates

b ~ 2 rp

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Double-diffractive scattering

• Both nucleons disassociates

b < rp

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Proton-(anti)Proton Collisions• At “high” energies we are probing the nucleon structure

– “High” means Ebeam >> hc/rproton ~ 1 GeV (Ebeam=1TeV@FNAL, 7TeV@LHC)

– We are really doing parton–parton scattering (parton = quark, gluon)

• Look for scatterings with large momentum transfer, ends up in detector “central region” (large angles wrt beam direction)– Each parton has a momentum distribution – CM of hard scattering is not fixed

as in e+e

• CM of parton–parton system will be moving along z-axis with some boost

• This motivates studying boosts along z– What’s “left over” from the other

partons is called the “underlying event”

• If no hard scattering happens, can still have disassociation– Underlying event with no hard

scattering is called “minimum bias”

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“Total Cross-section”

• By far most of the processes in nucleon-nucleon scattering are described by:– (Total) ~ (scattering) + (single diffractive) + (double diffractive)

• This can be naively estimated….– ~ 4rp

2 ~ 100mb

• Total cross-section stuff is NOT the reason we do these experiments!

• Examples of “interesting” physics @ Tevatron (2 TeV)– W production and decay via lepton

• Br(W e) ~ 2nb• 1 in 5x107 collisions

– Z production and decay to lepton pairs• About 1/10 that of W to leptons

– Top quark production• (total) ~ 5pb• 1 in 2x1010 collisions

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Phase Space

• Relativistic invariant phase-space element:– Define pp/pp collision axis along z-axis:

– Coordinates p = (E,px,py,px) – Invariance with respect to boosts along z?

• 2 longitudinal components: E & pz (and dpz/E) NOT invariant

• 2 transverse components: px py, (and dpx, dpy) ARE invariant

• Boosts along z-axis– For convenience: define p where only 1 component is not Lorentz invariant

– Choose pT, m, as the “transverse” (invariant) coordinates

• pT psin() and is the azimuthal angle

– For 4th coordinate define “rapidity” (y)

• …How does it transform?

E

dpdpdp

E

pdd zyx

3

z

z

pE

pEy

ln2

1yEpz tanhor

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Boosts Along beam-axis

• Form a boost of velocity along z axis– pz (pz + E)

– E (E+ pz)

– Transform rapidity:

• Boosts along the beam axis with v= will change y by a constant yb

– (pT,y,,m) (pT,y+yb,,m) with y y+ yb , yb ln (1+) simple additive to rapidity

– Relationship between y, , and can be seen using pz = pcos() and p = E

b

z

z

zz

zz

z

z

yyy

ypE

pE

EppE

EppE

pE

pEy

1ln1

1ln

2

1

ln2

1ln

2

1

cos1

cos1ln

2

1

y costanh yor where is the CM boost

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Phase Space (cont)

• Transform phase space element d from (E,px,py,pz) to (pt, y, , m)

• Gives:

• Basic quantum mechanics: d = |M |2d– If |M |2 varies slowly with respect to rapidity, ddy will be ~constant in y– Origin of the “rapidity plateau” for the min bias and underlying event structure – Apply to jet fragmentation - particles should be uniform in rapidity wrt jet axis:

• We expect jet fragmentation to be function of momentum perpendicular to jet axis

• This is tested in detectors that have a magnetic field used to measure tracks

E

dp

E

p

pE

p

pE

Edp

p

E

E

y

p

ydpdy

z

z

z

z

zz

zzz

2222

ddpdpdp Tyx2

2

1

dyddpd T 2

2

1

E

dpdpdp

E

pdd zyx

3

&

z

z

pE

pEy

ln2

1using

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Transverse Energy and Momentum Definitions

• Transverse Momentum: momentum perpendicular to beam direction:

• Transverse Energy defined as the energy if pz was identically 0: ETE(pz=0)

• How does E and pz change with the boost along beam direction?

– Using and gives

– (remember boosts cause y y + yb)– Note that the sometimes used formula is not (strictly) correct! – But it’s close – more later….

22222222zTyxT pEmpmppE

yEpz tanh

yEE T cosh

or222yxT ppp sinppT

sinEET

costanh y cosppz

yEyEEpEE zT22222222 sechtanh then

or which also means yEp Tz sinh

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Invariant Mass M1,2 of 2 particles p1, p2

• Well defined:

• Switch to p=(pT,y,,m) (and do some algebra…)

• This gives– With T pT/ET

– Note:• For y 0 and 0, high momentum limit: M 0: angles “generate” mass• For 1 (m/p 0)

This is a useful formula when analyzing data…

coscosh22121

22

21

22,1 TTTT yEEmmM

212122

21

221

22,1 2 ppEEmmppM

yEE T coshwith and TTT Ep

coscosh221

22,1 yEEM TT

2111 sinhsinhcos2121212121

yyEEpppppppp TTTTzzyyxx

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Invariant Mass, multi particles• Extend to more than 2 particles:

• In the high energy limit as m/p 0 for each particle:

Multi-particle invariant masses where each mass is negligible – no need to id Example: t Wb and W jet+jet

– Find M(jet,jet,b) by just adding the 3 2-body invariant masses

– Doesn’t matter which one you call the b-jet and which the “other” jets as long as you are in the high energy limit

23

22

21

23,2

23,1

22,1

23

23

22

2332

22

23

21

2331

21

22,1

233231

22,1

23321

221

2321

23,2,1

22

22

2

mmmMMM

mmmppppmmppppM

mppppM

mppppppppM

23,1

23,2

22,1

23,2,1 MMMM

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Pseudo-rapidity

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“Pseudo”rapidity and “Real” rapidity

• Definition of y: tanh(y) = cos()– Can almost (but not quite) associate position in the detector () with rapidity (y)

• But…at Tevatron and LHC, most particles in the detector (>90%) are ’s with 1

• Define “pseudo-rapidity” defined as y(,), or tanh() = cos() or

2tanln2sin

2cosln

cos1

cos1ln

2

1

=5, =0.77°)

CMS ECAL

CMS HCAL

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vs y

• From tanh() = cos() = tanh(y)/– We see that y – Processes “flat” in rapidity y will not be “flat” in pseudo-rapidity

-4

-3

-2

-1

0

1

2

3

4

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y

Beta=.5

Beta=.85

Beta=.99

Beta=.995

1.4 GeV

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– y and pT – Calorimater Cells

• At colliders, cm can be moving with respect to detector frame• Lots of longitudinal momentum can escape down beam pipe

– But transverse momentum pT is conserved in the detector

• Plot y for constant m, pT ()

• For all in DØ/CDF, can use position to give y:– Pions: |||y| < 0.1 for pT > 0.1GeV

– Protons: |||y| < 0.1 for pT > 2.0GeV

– As →1, y→ (so much for “pseudo”)

DØ calorimeter cell width

=0.1

pT=0.1GeV

pT=0.2GeV

pT=0.3GeV

CMS HCAL cell width 0.08

CMS ECAL cell width 0.005

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Rapidity “plateau”

• Constant pt, rapidity plateau means d/dy ~ k

– How does that translate into d/d ?

– Calculate dy/d keeping m, and pt constant

– After much algebra… dy/d =

– “pseudo-rapidity” plateau…only for 1

)tanh()()tanh( y

d

dyk

d

dy

dy

d

d

d

kd

dyk

d

dy

dy

d

d

d

222222

22

cosh

)cosh()(

TZT

ZT

pmmpp

pp

E

p

…some useful formulae…

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Transverse Mass

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Measured momentum conservation

• Momentum conservation: and

• What we measure using the calorimeter: and

• For processes with high energy neutrinos in the final state:

• We “measure” p by “missing pT” method:– e.g. W e or

• Longitudinal momentum of neutrino cannot be reliably estimated– “Missing” measured longitudinal momentum also due to CM energy going down beam

pipe due to the other (underlying) particles in the event– This gets a lot worse at LHC where there are multiple pp interactions per crossing

• Most of the interactions don’t involve hard scattering so it looks like a busier underlying event

cells

TT Epp

particles

CMZ Pp

cells

CMZ Pp

cells

TT pp 0

particles

Tp 0

cells

Tp 0

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Transverse Mass

• Since we don’t measure pz of neutrino, cannot construct invariant mass of W

• What measurements/constraints do we have?– Electron 4-vector

– Neutrino 2-d momentum (pT) and m=0

• So construct “transverse mass” MT by:

1. Form “transverse” 4-momentum by ignoring pz (or set pz=0)

2. Form “transverse mass” from these 4-vectors:

• This is equivalent to setting 1=2=0

• For e/ and , set me= m = m = 0 to get:

2121

22,1 TTTTT ppppM

0,, TTT pEp

)2(sin4cos12 222,1 2121

TTTTT EEEEM

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Transverse Mass Kinematics for Ws

• Transverse mass distribution?• Start with

• Constrain to MW=80GeV and pT(W)=0

– cos= -1

– ETe = ET

– This gives you ETeET versus

• Now construct transverse mass

– Cleary MT=MW when =0

coscosh22

,2

TTeW EEMMe

1cosh2

802

TTeEE

1cosh

802

cos122

2,

TTeTe EEM

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Neutrino Rapidity

• Can you constrain M(e,) to determine the pseudo-rapidity of the ?– Would be nice, then you could veto on in “crack” regions

• Use M(e,) = 80GeV and

• Since we know e, we know that =e ± – Two solutions. Neutrino can be either higher or lower in rapidity than electron

– Why? Because invariant mass involves the opening angle between particles.

– Clean up sample of W’s by requiring both solutions are away from gaps?

coscosh28022TTeW EEM

to get

cos2

80cosh

2

TTeEE

and solve for :

2

1coshcoshln

2

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Jets

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Jet Definition• How to define a “jet” using calorimeter towers so that we can use it for invariant mass calculations

– And for inclusive QCD measurements (e.g. d/dET)

• QCD motivated: – Leading parton radiates gluons uniformly distributed azimuthally around jet axis– Assume zero-mass particles using calorimeter towers

• 1 particle per tower– Each “particle” will have an energy perpendicular to the jet axis:– From energy conservation we expect total energy perpendicular to the jet axis to be zero on average:

– Find jet axis that minimizes kT relative to that axis– Use this to define jet 4-vector from calorimeter towers– Since calorimeter towers measure total energy, make a basic assumption:

• Energy of tower is from a single particle with that energy• Assume zero mass particle (assume it’s a pion and you will be right >90%!)• Momentum of the particle is then given by

– Note: mi=0 does NOT mean Mjet=0• Mass of jet is determined by opening angle between all contributors• Can see this in case of 2 “massless” particles, or energy in only 2 towers:

• Mass is “generated” by opening angles. • A rule of thumb: Zero mass parents of decay have 12=0 always

Tk

2sin4)cos1(2 122

21122112

EEEEM

0particles

Tk

and points to tower i with energy iii nEp ˆ

in̂ iE

iE

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Quasi-analytical approach

• Transform each calorimeter tower to frame of jet and minimize kT

– 2-d Euler rotation (in picture, =jet, =jet, set =0)

– Tower in jet momentum frame: and apply

– Check: for 1 tower, tower=jet, should get Exi = Exi = 0 and Ezi = Ejet

• It does, after some algebra…

jetjetjetjetjet

jetjetjetjetjet

jetjet

jetjetM

cossinsincossin

sinsincoscoscos

0cossin

,

ijetjeti EME , 0particles

Tk

jetzijetjetyijetjetxizi

jetzijetjetyijetjetxiyi

jetyijetxixi

EEEE

EEEE

EEE

cossinsincossin

sinsincoscoscos

cossin

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Minimize kT to Find Jet Axis

• The equation is equivalent to so…

• Momentum of the jet is such that:

0particles

Tk 0 i

yii

xi EE

0cossin yijetxijetxi EEE

xi

yijet E

Etan

0sinsincoscos zijetyijetxijetjetyi EEEE

zi

yixi

jet E

EE22

tan

jetx

jetyjet p

p

,

,tan

yijety

xijetx

Ep

Ep

,

,

zijetz

yixijetT

Ep

EEp

,

22

,

jetz

jetTjet p

p

,

,tan

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Jet 4-momentum summary

• Jet Energy:

• Jet Momentum:

• Jet Mass:

• Jet 4-vector:

• Jet is an object now! So how do we define ET?

cellsi

cellsijetjet

jet pEpEp

,,

towers

iijet nEp ˆ

towers

ijet EE

222jetjetjet pEM

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ET of a Jet

• For any object, ET is well defined:

• There are 2 more ways you could imagine using to define ET of a jet but neither are technically correct:

– How do they compare?

– Is there any ET or dependence?

jetjetjetT EE sin, towers

iTjetT EE ,,

22,

2,

2, jetjetTjetzjetjetT mppEE

or

correct

Alternative 1

Alternative 2

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True ET vs Alternative 1

• True:

• Alternative 1:

• Define which is always >0

– Expand in powers of :

– For small , tanh so either way is fine• Alternative 1 is the equivalent to true def central jets

– Agree at few% level for ||<0.5

– For ~0.5 or greater....cone dependent• Or “mass” dependent....same thing

22,, jetjetTjetT mpE

22,

222,

,

,1

sin1

sin

jetjetT

jetjetjetT

jetT

jetjetjetT

mp

mp

E

EE

2,

2

jetT

jet

p

m2

,

22

1 2

tanh

jetT

jetjet

p

m

jetjetjetjetjetjetjetjetjetT mpmpEE 22222, sinsinsin

Leading jet, ||>0.5

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True ET vs Alternative 2

Alternative 2: harder to see analytically…imagine a jet w/2 towers

– TRUE:

– Alternative 2:

– Take difference:

– So this method also underestimates “true” ET

• But not as much as Alternative 1

towers

iTjetT EE ,,

21212

221

2121

21

2121

21

221

221

2,

22,

coscos12

22

EEEE

ppppEEEE

ppEEpEE

TT

zzz

zzjetzjetjetT

2121

22

21

212

221

211

sinsin2

2

EEEE

EEEEEE

TT

TTTTTT

2sincos12

sinsincoscos122

2121

2121212

212

,

EEEE

EEEEE TTjetT

Always > 0!

22,

2 1jetjetT

Ti

mp

E

Leading jet, ||>0.5

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Jet Shape

• Jets are defined by but the “shape” is determined by

• From Euler:

• Now form for those towers close to the jet axis: 0 and 0

• From we get which means

0, particles

iTk

02,

2,

2,

particlesiyix

particlesiT EEk

jetzijetiTi

jetzijetjetyijetjetxiyi

iTijetyijetxixi

EE

EEEE

EEEE

sincoscos

sinsincoscoscos

sincossin

particles

iTk2

,

iiiijetiijetiijetzijetTiyi

iTixi

EEEEEEE

EE

~sinsincoscossinsincos

jeti

jeti

costanh dd sin

iTiiiiiiyi

iTixi

EEEE

EE

sin

222,

222, iiiTyixiiT EEEk So…

and…

particlesiiiT

particlesiyix

particlesiT EEEk 222

,2,

2,

2,

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Jet Shape – ET Weighted

• Define and

– This gives: and equivalently,

– Momentum of each “cell” perpendicular to jet momentum is from• Eti of particle in the detector, and

• Distance from jet in plane

– This also suggests jet shape should be roughly circular in plane• Providing above approximations are indicative overall….

• Shape defined:– Use energy weighting to calculate true 2nd moment in plane

particles

iiTparticles

iT REk 22,

2,

222iiiR

iTiTi REk

222iiii RR

particlesiT

particlesiiT

particlesiT

particlesiT

R E

RE

E

k

2,

22,

2,

2,

2

particlesiT

particlesiiT

E

E

2,

22,

particlesiT

particlesiiT

E

E

2,

22,

with

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Jet Shape – ET Weighted (cont)

• Use sample of “unmerged” jets

• Plot

– Shape depends on cone parameter– Mean and widths scale linearly with cone

parameter

particlesiT

particlesiyix

R E

EE

2,

2,

2,

• “Small angle” approximation pretty good

For Cone=0.7, distribution in R has:

Mean ± Width =.25 ± .05

99% of jets have R <0.4

<R> vs Jet Clustering Parameter (Cone Size)

< R

>

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Jet Mass

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Jet Samples

• Run 1• All pathologies eliminated (Main Ring, Hot Cells, etc.)

• |Zvtx|<60cm

• No , e, or candidates in event– Checked coords of e vs. jet list

– Cut on cone size for jets• .025, .040, .060 for jets from cone cuttoff 0.3, 0.5, 0.7 respectively

• “UNMERGED” Sample:– RECO events had 2 and only 2 jets for cones .3, .5, and .7

– Bias against merged jets but they can still be there• e.g. if merging for all cones

• “MERGED” Sample:– Jet algorithm reports merging

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Jet Mass

• Jet is a physics object, so mass is calculated using:– Either one…

• Note: there is no such thing as “transverse mass” for a jet– Transverse mass is only defined for pairs (or more) of 4-vectors…

• For large ET,jet we can see what happens by writing

– And take limit as jet narrows and and expand ET and pT

– This gives

so…. using

2,

2,

222jetTjetTjetjetjet pEpEM 2

,2

,222

jetTjetTjetjetjet pEpEM

jetTjetTjetTjetTjetTjetTjet pEpEpEM ,,,,2

,2

,2

0i

21

2

,,i

iTjetT Ep

21

2

,,i

iTjetT EE

0i

22,,, 2

1iiiTjetTjetT EpE iTiiiTjetTjetT EEpE ,

22,,, 24

2

1

222iiTiTijet EEM RjetTjet EM ,

particlesiTjetT EE ,,

Jet mass is related to jet shape!!! (in the thin jet, high energy limit)

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Jet Mass (cont)

• Jet Mass for unmerged sample How good is “thin jet” approximation?

Low-side tail is due to lower ET jets for smaller cones

(this sample has 2 and only 2 jets for all cones)

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Jet Merging

• Does jet merging matter for physics? – For some inclusive QCD studies, it doesn’t matter

– For invariant mass calculations from e.g. topWb, it will smear out mass distribution if merging two “tree-level” jets that happen to be close

• Study R…see clear correlation between R and whether jet is merged or not

– Can this be used to construct some kind of likelihood?“Unmerged”, Jet Algorithm reports merging, all cone sizes “Unmerged” v. “Merged” sample

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Merging Likelihood

• Crude attempt at a likelihood– Can see that for this (biased) sample, can use this to pick out “unmerged” jets

based on shape

– Might be useful in Higgs search for H bb jet invariant mass?

Jet cone parameter

Equal likelihood to be merged and unmerged

0.3 0.155

0.5 0.244

0.7 0.292

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Merged Shape

• Width in “assumes” circular

– Large deviations due to merging?

– Define should be independent of cone size

• Clear broadening seen – “cigar”-shaped jets, maybe study…

2R

“Unmerged” Sample “Merged” Sample

particlesiT

particlesiiiT

E

E

2,

2,