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2 - Collider Physics. 2.1 Phase space and rapidity - the “plateau” 2.2 Source Functions - protons to partons 2.3 Pointlike scattering of partons 2.4 2-->2 formation kinematics 2.5 2--1 Drell-Yan processes 2.6 2-->2 decay kinematics - “back to back” 2.7 Jet Fragmentation. - PowerPoint PPT Presentation
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FNAL Academic Lectures – May, 2006 1
2 - Collider Physics2 - Collider Physics 2 - Collider Physics2 - Collider Physics
• 2.1 Phase space and rapidity - the “plateau”
• 2.2 Source Functions - protons to partons
• 2.3 Pointlike scattering of partons
• 2.4 2-->2 formation kinematics
• 2.5 2--1 Drell-Yan processes
• 2.6 2-->2 decay kinematics - “back to back”
• 2.7 Jet Fragmentation
FNAL Academic Lectures – May, 2006 2
Kinematics - RapidityKinematics - RapidityKinematics - RapidityKinematics - Rapidity
One Body Phase Space
NR
ddPPdPdPdPPd TT||2
EPdmPPd /224
2
TPdydEdPdy /||
Relativistic
6.9,7.7
,14,2@
,0max
cosh
max
222
y
TeVpp
momentumbeamPaty
Pmm
ymE
T
TT
T
Rapidity
If transverse momentum is limited by dynamics, expect a uniform distribution in y
Kinematically allowed range in y of a proton with PT=0
FNAL Academic Lectures – May, 2006 3
Rapidity “Plateau”Rapidity “Plateau”Rapidity “Plateau”Rapidity “Plateau”
Monte Carlo results are homebuilt or COMPHEP - running under Windows or Linux
Region around y=0 (90 degrees) has a “plateau” with width y ~ 6 for LHC
LHC
FNAL Academic Lectures – May, 2006 4
Rapidity Plateau - JetsRapidity Plateau - JetsRapidity Plateau - JetsRapidity Plateau - Jets
For ET small w.r.t sqrt(s) there is a rapidity plateau at the Tevatron with y ~ 2 at ET < 100 GeV.
FNAL Academic Lectures – May, 2006 5
Parton and Hadron DynamicsParton and Hadron DynamicsParton and Hadron DynamicsParton and Hadron DynamicsFor large ET, or short distances, the impulse approximation means that quantum effects can be ignored. The proton can be treated as containing partons defined by distribution functions. f(x) is the probability distribution to find a parton with momentum fraction x.
Proceed left to right
FNAL Academic Lectures – May, 2006 6
The “Underlying Event”The “Underlying Event”The “Underlying Event”The “Underlying Event”
The residual fragments of the pp resolve into soft - PT ~ 0.5 GeV pions with a density ~ 5 per unit of rapidity (Tevatron) and equal numbers of +o-. At higher PT, “minijets” become a prominent feature
2.8~,3.1~,/450~
)/(~/2
2
nGeVpGeVmbA
ppAdydpd
o
noTT
s dependence for PT < 5 GeV is small
FNAL Academic Lectures – May, 2006 7
COMPHEP - Minijets COMPHEP - Minijets COMPHEP - Minijets COMPHEP - Minijets
p-p at 14 TeV, subprocess g+g->g+g, cut on Ptg> 5 GeV. Note scale is mb/GeV
FNAL Academic Lectures – May, 2006 8
Minijets - Power Law?Minijets - Power Law?Minijets - Power Law?Minijets - Power Law?
The very low PT fragments change to “minijets” - jets at “low” PT which have mb cross sections at ~ 10 GeV. The boundary between “soft, log(s)” physics and “hard scattering” is not very definite. Note log-log, which is not available in COMPHEP – must export the histogram
pp(g+g) -> g + g
FNAL Academic Lectures – May, 2006 9
The Distribution FunctionsThe Distribution FunctionsThe Distribution FunctionsThe Distribution Functions
•Suppose there was very weak binding of the u+u+d “valence” quarks in the proton.
•But quarks are bound, .
•Since the quark masses are small the system is relativistic - “valence” quarks can radiate gluons ==> xg(x) ~ constant. Gluons can “decay” into pairs ==> xs(x) ~ constant. The distribution is, in principle, calcuable but not perturbatively. In practice measure in lepton-proton scattering.
x ~ 1/3, f(x) is a delta function
~ , ~ 1 , ~ 0.2 ~x x QCDx P x fm P GeV
FNAL Academic Lectures – May, 2006 10
Radiation - Soft and CollinearRadiation - Soft and CollinearRadiation - Soft and CollinearRadiation - Soft and Collinear
P (1-z)P
z
PzPE
EEE
PmPmPE
if
/1~
)1(~
)/(1/1~
2/222
,k
cosk
kPP
EE
The amplitude for radiation of a gluon of momentum fraction z goes as ~ 1/z. The radiated gluon will be ~ collinear - ~ k ==> ~ 0. Thus, radiated objects are soft and collinear.
Cherenkov relation
FNAL Academic Lectures – May, 2006 11
COMPHEP, e+t->e+t+ACOMPHEP, e+t->e+t+ACOMPHEP, e+t->e+t+ACOMPHEP, e+t->e+t+A
Use heavy quark as a source of photons – needed to balance E,P. See strong forward (electron-photon) peak.
FNAL Academic Lectures – May, 2006 12
Parton Distribution FunctionsParton Distribution FunctionsParton Distribution FunctionsParton Distribution Functions
)1(~)(
2/1)(
)1(2/7)( 6
xxfx
dxxxg
xxxg
“valence” “sea” gluons
In the proton, u and d quarks have largest probability at large x. Gluons and “sea” anti-quarks have large probability at low x. Gluons carry ~ 1/2 the proton momentum. Distributions depend on distance scale (ignore).
FNAL Academic Lectures – May, 2006 13
Proton – Parton Density Proton – Parton Density FunctionsFunctions
Proton – Parton Density Proton – Parton Density FunctionsFunctions
g dominates for x < 0.2
At large x, x > 0.2, u dominates over d and g.
“sea” dominates for x < 0.03 over valence.
Points are simple xg(x) parametrization.
FNAL Academic Lectures – May, 2006 14
2-->2 Formation Kinematics2-->2 Formation Kinematics2-->2 Formation Kinematics2-->2 Formation Kinematics
tan/1sinh
sin/1cosh
:0
/,/
)/ln(2~,)/(~,sinh
cosh
/0
,/
,/2
222
||
21
212
21
||
y
y
m
EPmE
Pmm
MsyesMxymP
ymE
sMxxx
xxxsMxx
sPx
TT
yT
T
E.g. for top quark pairs at the Tevatron, M ~ 2Mt ~ 350 GeV. <x> ~ ~350/1800 ~ 0.2
Top pairs produced by quarks.
x1 x2
2
2 2 2 21 2 1 2
~ 4
~ [( ) ( ) ]
s P
M P x x x x
FNAL Academic Lectures – May, 2006 15
Linux COMPHEPLinux COMPHEPLinux COMPHEPLinux COMPHEP
g + g->g + g with Pt of final state gluons > 50 GeV at 14 TeV p-p
n.b. To delete diagrams use d, o to turn them back on one at a time
Cross section is 0.013 mb (very large)
Write out full events – but no fragmentation. COMPHEP does not know about hadrons.
FNAL Academic Lectures – May, 2006 16
gg -> gg in Linux COMPHEPgg -> gg in Linux COMPHEPgg -> gg in Linux COMPHEPgg -> gg in Linux COMPHEP
Note the kinematic boundary, where <x> ~ 0.007 is the y=0 value for x1=x2 for M = 100, C.M. = 14000.
FNAL Academic Lectures – May, 2006 17
CDF Data – DY Electron PairsCDF Data – DY Electron PairsCDF Data – DY Electron PairsCDF Data – DY Electron Pairs
DY Plateau
x1,x2 at Z mass ~ 0.045
FNAL Academic Lectures – May, 2006 18
The Fundamental Scattering The Fundamental Scattering AmplitudeAmplitude
The Fundamental Scattering The Fundamental Scattering AmplitudeAmplitude
.| | ~ ( )iq rI IA f H i e V r dr
Fourier transform of the interaction potential, VI(r) where , ~f iq k k q k
is the
magnitude of the momentum transfer in the reaction. A familiar example is the 1/r Coulomb
potential, which yields a Born amplitude ~ 1/q2 describing how the virtual exchanged photon
propagates in momentum space. In turn this leads to a cross section (Rutherford scattering)
which goes as the square of the amplitude ~ 1/q4~ 1/4 , which should be familiar.
1 1 2
2 3 4
2 2
~
~
[ ] [ ] [1/ ] 1/
x x x vertex
xx x vertex
L M s
FNAL Academic Lectures – May, 2006 19
Pointlike Parton Cross SectionsPointlike Parton Cross SectionsPointlike Parton Cross SectionsPointlike Parton Cross Sections
Point-like cross sections for parton - parton scattering. The entries have the generic dependence already factored out. At large transverse momenta, or scattering angles near 90 degrees (y ~ 0),
the remaining factors are dimensionless numbers of order one.
Process 2A Value at
q q q q 2 2 24[ ] /
9s u t
2.22
q q q q 2 2 2 2 2 2 24 8
[( ) / ( ) / ] ( / )9 27
s u t s t u s ut 3.26
q q q q 2 2 24[ ] /
9t u s
0.22
q q q q 2 2 2 2 2 2 24 8[( ) / ( ) / ] ( / )
9 27s u t t u s u st
2.59
q q g g 2 2 2 2 232 8[ ] / [ ] /
27 3t u tu t u s
1.04
g g q q 2 2 2 2 21 3[ ] / [ ] /
6 8t u tu t u s
0.15
g q g q 2 2 2 2 24
[ ] / [ ] /9
s u su u s t 6.11
g g g g 2 2 29
[3 / / / ]2
tu s su t st u 30.4
q q g 2 28[ ] /
9t u tu
g q q 2 21
[ ] /3
s u su
Pointlike partons have Rutherford like behavior
~ (12)|A|2/s
s,t,u are Mandelstam variables. |A|2 ~ 1 at y=0.
FNAL Academic Lectures – May, 2006 20
Hadronic Cross SectionsHadronic Cross SectionsHadronic Cross SectionsHadronic Cross Sections
)4321(ˆ)()(/
)4321(ˆ)()(
//ˆ
/ˆ
)4321(ˆ)()(ˆ
210
2211
2
21
212211
dfCfdydd
dyddxfxCfd
sMss
dydsdysddxdx
ddxdxxfxCfdPPd
y
BA
To form the system need x1 from A and x2 from B picked out of probability distributions with the joint probability PAPB to form a system of mass M moving with momentum fraction x. C is a color factor (later). The cross section is ~ (d/dy)y=0y. The value of y varies only slowly with mass ~ ln(1/M)
FNAL Academic Lectures – May, 2006 21
2-->2 and 2-->1 Cross Sections2-->2 and 2-->1 Cross Sections2-->2 and 2-->1 Cross Sections2-->2 and 2-->1 Cross Sections
31 20
1 2
4 20 1 2 1 2
2
2
21 20
2 2 :
ˆ ˆ/ 2 ( ) ( )
ˆ ˆ/
( / ) ~ [ ( ) ( )] ( )
2 1:
ˆ 4 (2 1),
ˆ ˆ (2 1)( / ), int
/ ( ) ( )
y x
y x
y x
M d dydM C xf x xf x d s
d s
M d dydM C xf x xf x
J partial wave unitarity
ds J M egrate over narrow width
M d dy C xf x xf x
2
12
2 21 2 120
(2 1) /
/ ~ " "
/ ( ) ( ) (2 1)
ff
y x
J M
M
M d dy C xf x xf x J
“scaling” behavior – depends only on and not M and s separately
FNAL Academic Lectures – May, 2006 22
DY Formation: 2 --> 1DY Formation: 2 --> 1DY Formation: 2 --> 1DY Formation: 2 --> 1
At a fixed resonant mass, expect rapid rise from “threshold” - ~
(1-M/s)2a
- then slow “saturation”. W ~ 30 nb at the LHC
, eu u Z e e u d W e
FNAL Academic Lectures – May, 2006 23
DY Z Production – F/B AsymmetryDY Z Production – F/B AsymmetryDY Z Production – F/B AsymmetryDY Z Production – F/B AsymmetryCDF – Run I
The Z couples to L and R quarks differently -> parity violating asymmetry in the photon-Z interference.
FNAL Academic Lectures – May, 2006 24
F/B AsymmetryF/B AsymmetryF/B AsymmetryF/B Asymmetry
3, / 2, 1, 2
, , 1/ 3, 4 / 3, 2 / 3
R L R
L
R R L uR dR
L
e Q I Y Y Ye
uu d Y Y Y
d
23/ cos [ sin ]W W Wg I Q
Coupling of leptons and quarks to Z specified in SM by gauge principle.
Coupling to L and R fermions differs => P violation ~ R-L coupling. Predict asymmetry , A ~ I3/Q. Thus, A for muons = 1, that for u quarks is 3/2, while for d quarks it is 3.
FNAL Academic Lectures – May, 2006 25
COMPHEPCOMPHEPCOMPHEPCOMPHEP
At 500 GeV the asymmetry is large and positive – here not p-p but u-U
FNAL Academic Lectures – May, 2006 26
COMPHEP - AssymCOMPHEP - AssymCOMPHEP - AssymCOMPHEP - Assym
Option in “Simpson” to get F/B asymmetry in COMPHEP
FNAL Academic Lectures – May, 2006 27
DY Formation of CharmoniumDY Formation of CharmoniumDY Formation of CharmoniumDY Formation of Charmonium
Cross section = ~ 2(2J+1)/M3 for W, width ~ 2 GeV, = 47 nb. For charmonium, width is 0.000087 GeV, and estimate cross section in gg formation as 34 nb. The PT arises from ISR and intrinsic parton transverse momentum and is only a few GeV, on average. Use for lepton momentum scale and resolution.
g
g
FNAL Academic Lectures – May, 2006 28
Charmonium CalibrationCharmonium CalibrationCharmonium CalibrationCharmonium Calibration
Cross section in |y|<1.5 is ~ 800 nb at the LHC. Lepton calibration – mass scale, width?
FNAL Academic Lectures – May, 2006 29
Upsilon CalibrationUpsilon CalibrationUpsilon CalibrationUpsilon Calibration
Cross section * BR about 2 nb at the LHC. Resolve the spectral peaks? Mass scale correct?
FNAL Academic Lectures – May, 2006 30
ZZ Production vs CM EnergyZZ Production vs CM EnergyZZ Production vs CM EnergyZZ Production vs CM Energy
VV production also has a steep rise near threshold. There is a 20 fold rise from the Tevatron to the LHC. Measure VVV coupling. ZZ has ~ 2 pb cross section at LHC.
Not much gain in using anti-protons once the energy is high enough that the gluons or “sea” quarks dominate.
FNAL Academic Lectures – May, 2006 31
WWZ – Quartic CouplingWWZ – Quartic CouplingWWZ – Quartic CouplingWWZ – Quartic Coupling
Not accessible at Tevatron. Test quartic couplings at the LHC.
FNAL Academic Lectures – May, 2006 32
Jet-Jet Mass, 2 --> 2Jet-Jet Mass, 2 --> 2Jet-Jet Mass, 2 --> 2Jet-Jet Mass, 2 --> 2
Expect 1/M3 behavior at low mass. When M/s becomes substantial, the source effects will be large. E.g. for M = 400 GeV, at the Tevatron, M/s=0.2, and
(1-M/s)12 is ~ 0.07. 3 12/ ~ (1 / )
p p g g
M d dM M s
FNAL Academic Lectures – May, 2006 33
Jets - 2 TeV- |y|<2Jets - 2 TeV- |y|<2Jets - 2 TeV- |y|<2Jets - 2 TeV- |y|<2
ET ~ M/2 for large scattering angles.
1/M3[1-M/s]12
behavior
FNAL Academic Lectures – May, 2006 34
COMPHEP LinuxCOMPHEP LinuxCOMPHEP LinuxCOMPHEP Linux
/ 2TP M
FNAL Academic Lectures – May, 2006 35
Scaling ?Scaling ?Scaling ?Scaling ?
Tevatron runs at 630 and 1800 GeV in Run I. Test of scaling in inclusive jet production. Expect a function of
only in lowest order.
2 /T Tx P s
FNAL Academic Lectures – May, 2006 36
Direct Photon ProductionDirect Photon ProductionDirect Photon ProductionDirect Photon Production
Expect a similar spectrum with a rate down by ratio of coupling constants and differences in u and g source functions. /s~14
u/g~6 at x~0.
FNAL Academic Lectures – May, 2006 37
D0 Single PhotonD0 Single PhotonD0 Single PhotonD0 Single Photon
Process dominated by q + g – a la Compton scattering.
COMPHEP – 2 TeV p-p
FNAL Academic Lectures – May, 2006 38
2--> 2 Kinematics - “Decays”2--> 2 Kinematics - “Decays”2--> 2 Kinematics - “Decays”2--> 2 Kinematics - “Decays”
y
y
esMx
yyyesMx
]/[
2/)(,]/[
2
431
x1 x2 x,y,M y3, y4 y*, *
Formation System Decay CM Decay
The measured values of y3, y4
and ET allow one to solve for the initial state x1 and x2 and the c.m. decay angle.
3 4ˆ ( ) / 2
ˆ ˆcos tanh( )
y y y
y
FNAL Academic Lectures – May, 2006 39
COMPHEP - LinuxCOMPHEP - LinuxCOMPHEP - LinuxCOMPHEP - Linux
g+g-> g+ g, in pp at 14 TeV with cut of Pt of jets of 50 GeV.
See a plateau for jets and the t channel peaking. Want to establish jet cross section, angular distributions and to look at jet “balance” – missing Et distribution in dijet events. MET angle ~ jet azimuthal angle and no non-Gaussian tails.
FNAL Academic Lectures – May, 2006 40
Parton-->Hadron FragmentationParton-->Hadron FragmentationParton-->Hadron FragmentationParton-->Hadron Fragmentation
)/ln(~~
/~/
)/ln(~/~)(
)1()(
/,1
/
||
1
/
minmin
Msyn
zdzEdPdy
mPazdzadzzDn
zazzD
Pmzzz
Pkz
Pm
For light hadrons
(pions) as hadronization products, assume kT is limited (scale ~. The fragmentation function, D(z) has a radiative form, leading to a jet multiplicity which is logarithmic in ET
Plateau widens with s, <n>~ln(s)
FNAL Academic Lectures – May, 2006 41
CDF Analysis – Jet MultiplicityCDF Analysis – Jet MultiplicityCDF Analysis – Jet MultiplicityCDF Analysis – Jet Multiplicity
Different
Cone radii
Jet cluster multiplicity within a cone increases with dijet mass as ~ ln(M).
FNAL Academic Lectures – May, 2006 42
Jet Transverse ShapeJet Transverse ShapeJet Transverse ShapeJet Transverse Shape
There is a “leading fragment” core localized at small R w.r.t. the jet axis - 40% of the energy for R< 0.1. 80% is contained in R < 0.4 cone 22 yR
FNAL Academic Lectures – May, 2006 43
Jet Shape - Monte CarloJet Shape - Monte CarloJet Shape - Monte CarloJet Shape - Monte Carlo
Simple model with zD(z) ~ (1-z)5 and <kt> ~ 0.72 GeV. “Leading fragment” with <zmax> ~ 0.24. On average the leading fragment takes ~ 1/4 of the jet momentum. Fragmentation is soft and non-perturbative.
FNAL Academic Lectures – May, 2006 44
Low Mass LHC RatesLow Mass LHC RatesLow Mass LHC RatesLow Mass LHC Rates
2 2 27 2
3 2 20
2 2 2 2
2
" " :
( ) 0.4 , 1 10
( / ) 2[ ( )] ( )( )
( ) ~ [ ( )] [ | | / ]
( ) ~ 7 / 2, ~ 10, ~ 0.1, ~ 1
10 , | | 30 ( )
~ 0.4
y
o s o
s
o
Minijet Rate
c mbGeV mb cm
M d dMdy xg x C d s c
M M y xg x A M
xg x y C
for M GeV A gg gg
mb
34 2
9
Re :
~ 100
~ 10 /( sec)
~ 25 sec
~ 10
~ 25 min / sinx
Total actionRate
mb
L cm
t n
L Hz
n bias events cros g
For small x and strong production, the cross section is a large fraction of the inelastic cross section. Therefore, the probability to find a “small Pt “minijet” in an LHC crossing is not small.
FNAL Academic Lectures – May, 2006 45
V V Production - W + V V Production - W + V V Production - W + V V Production - W +
The angular distribution at the parton level has a zero. The SM prediction could be confirmed with a large enough event sample. – pp at 2 TeV with Pt > 10 GeV, 0.6 pb
Asymmetry somewhat washed out by the contribution of sea anti-quarks in the p and sea quarks in the anti-proton.