jets, kinematics, and other variables
DESCRIPTION
Jets, Kinematics, and Other Variables. A Tutorial Drew Baden University of Maryland World Scientific Int.J.Mod.Phys.A13:1817-1845,1998. Detector. x. f. q. r. Proton beam direction. Proton or anti-proton beam direction. z. Protons. Anti-Protons. Transverse E E T. y. E. - PowerPoint PPT PresentationTRANSCRIPT
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
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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,
18-Jan-2006 UCLA 33
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
18-Jan-2006 UCLA 41
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,