Collider Physics

1
Collider Physics

• 3.1 Phase space and rapidity - the plateau
• 3.2 Source Functions - protons to partons
• 3.3 Pointlike scattering of partons
• 3.4 2--gt2 formation kinematics
• 3.5 2--1 Drell-Yan processes
• 3.6 2--gt2 decay kinematics - back to back
• 3.7 Jet Fragmentation

2
Kinematics - Rapidity

• One Body Phase Space
• NR

Rapidity
Relativistic
If transverse momentum is limited by dynamics,
expect a uniform distribution in y
Kinematically allowed range in y of a proton with
PT0
3
Rapidity Plateau
Monte Carlo results are homebuilt or COMPHEP -
running under Windows or Linux
Region around y0 (90 degrees) has a plateau
with width ?y 6 for LHC


LHC
4
Rapidity Plateau - Jets
For ET small w.r.t sqrt(s) there is a rapidity
plateau at the Tevatron with ?y 2 at ET lt 100
GeV.
5
Parton and Hadron Dynamics
For 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.
6
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
s dependence for PT lt 5 GeV is small
7
Minijets
pp(gg) -gt g g
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.
8
The Distribution Functions

• Suppose there was very weak binding of the uud
  valence quarks in the proton.
• But quarks are bound, .
• Since the quark masses are small the system is
  relativistic - valence quarks can radiate
  gluons gt xg(x) constant. Gluons can decay
  into pairs gt 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
9
Radiation - Soft and Collinear
?,k
The amplitude for radiation of a gluon of
momentum fraction z goes as 1/z. The radiated
gluon will be collinear - ? k gt ? 0.
Thus, radiated objects are soft and collinear.
P (1-z)P
Cherenkov relation
10
Parton Distribution Functions
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).
valence sea gluons
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Proton Parton Density Functions
g dominates for x lt 0.2 At large x, u dominates
over d sea dominates for x lt 0.03
Points are simple xg(x) parametrization.
12
2--gt2 Formation Kinematics
E.g. for top quark pairs at the Tevatron, M 2Mt
350 GeV. ltxgt ??350/1800 0.2
x1
x2
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2 --gt 2 Production
Simple Monte Carlo picks x1 from g(x) and x2 out
of g(x), weighting by the dynamics, 1/M2. Note
the kinematic boundary, where ltxgt 0.1 is the
y0 value for x1x2.
14
Linux Comphep

• gg-gtgg with Pt of final state gluons gt 50 GeV
• n.b. d to delete diagrams, 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

15
gg -gt gg in Linux Comphep
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CDF Data DY Electron Pairs
17
The Fundamental Scattering Amplitude
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Pointlike Parton Cross Sections
Pointlike partons have Rutherford like
behavior ? ?(?1?2)A2/s s,t,u are
Mandelstam variables. A2 1 at y0.
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Hadronic Cross Sections
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). On the plateau, the cross section
is ?? (d?/dy)y0?y. The value of ?y varies only
slowly with mass ln(1/M)
20
2--gt2 and 2--gt1 Cross Sections
scaling behavior depends only on ? and not M
and s separately
21
DY Formation 2 --gt 1
At a fixed resonant mass, expect rapid rise from
threshold - ?? (1-M/?s)2a - then slow
saturation. ?W 30 nb at the LHC
22
DY Z Production F/B Asymmetry
CDF Run I
The Z couples to L and R quarks differently -gt
parity violating asymmetry in the photon-Z
interference.
23
COMPHEP
At 500 GeV the asymmetry is large and positive
24
DY Formation of Charmonium
Cross section ? ?2?(2J1)/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 (PTW - Chpt. 4).
g
?
g
25
ZZ 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.
26
WWZ Quartic Coupling
27
Low Mass LHC Rates
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.
28
Jet-Jet Mass, 2 --gt 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/?s0.2, and (1-M/?s)12 is 0.07.
29
Jets - 2 TeV- ylt2


1/M31-M/?s12 behavior
ET M/2 for large scattering angles.
30
COMPHEP Linux
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Scaling ?
Tevatron runs at 630 and 1800 GeV in Run I. Test
os scaling in inclusive jet production. Expect a
function of
only in lowest order.
32
Direct Photon Production
Expect a similar spectrum with a rate down by
ratio of coupling constants and differences in u
and g source functions. ?/?s14 u/g6 at x0.
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2--gt 2 Kinematics - Decays

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.
34
2--gt2 Decay Distribution
35
Comphep - Linux
gg-gt 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.
36
2--gt2, x of System, y of Decay
Note that x is limited to be 0 by the limited x
values of the source functions
Note the plateau for decayproducts is limited
to ?y2 at the Tevatron at these masses
37
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 gt 10 GeV, 0.6 pb
38
Parton--gtHadron Fragmentation
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, ltngtln(s)
39
CDF Analysis Jet Multiplicity
Different Cone radii
Jet cluster multiplicity within a cone increases
with dijet mass as ln(M).
40
Jet Transverse Shape
There is a leading fragment core localized at
small R w.r.t. the jet axis - 40 of the energy
for Rlt 0.1. 80 is contained in R lt 0.4 cone
41
Jet Shape - Monte Carlo
Simple model with zD(z) (1-z)5 and ltktgt 0.72
GeV. Leading fragment with ltzmaxgt 0.24. On
average the leading fragment takes 1/4 of the
jet momentum. Fragmentation is soft and
non-perturbative.