2 - Collider Physics

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


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


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


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.


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


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


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


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


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


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


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.


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).


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.


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


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.


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.


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


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


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.


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)


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


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


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.


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.


COMPHEPCOMPHEPCOMPHEPCOMPHEP

At 500 GeV the asymmetry is large and positive – here not p-p but u-U


COMPHEP - AssymCOMPHEP - AssymCOMPHEP - AssymCOMPHEP - Assym

Option in “Simpson” to get F/B asymmetry in COMPHEP


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


Charmonium CalibrationCharmonium CalibrationCharmonium CalibrationCharmonium Calibration

Cross section in |y|<1.5 is ~ 800 nb at the LHC. Lepton calibration – mass scale, width?


Upsilon CalibrationUpsilon CalibrationUpsilon CalibrationUpsilon Calibration

Cross section * BR about 2 nb at the LHC. Resolve the spectral peaks? Mass scale correct?


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.


WWZ – Quartic CouplingWWZ – Quartic CouplingWWZ – Quartic CouplingWWZ – Quartic Coupling

Not accessible at Tevatron. Test quartic couplings at the LHC.


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


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


COMPHEP LinuxCOMPHEP LinuxCOMPHEP LinuxCOMPHEP Linux

/ 2TP M


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


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.


D0 Single PhotonD0 Single PhotonD0 Single PhotonD0 Single Photon

Process dominated by q + g – a la Compton scattering.

COMPHEP – 2 TeV p-p


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


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.


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)


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).


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


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.


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.


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.

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