UCT Seminar V: A Brief Intro to Glauber Calculations

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UCT Seminar VA Brief Intro to Glauber
Calculations

• Peter Steinberg
• Brookhaven National Laboratory
• Fulbright Scholar Program

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Why Geometry Matters

• Binary Collisions
• Jet Production
• Heavy Flavor

b
Glauber model of AA
Binary Collisions
Npart, Ncoll

• Color Exchange
• Soft Hadron Production
• Transverse Energy

Participants
wounded nucleon model
b (fm)
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Glauber calculations

• Nucleons are distributed according to a density
  function (e.g. Woods-Saxon)
• Nucleons travel in straight lines and are not
  deflected as they pass through the other nucleus
• Nucleons interact according to the inelastic
  cross section sNN measured in pp collisions, even
  after interacting
• Participants counts nucleons which interact
• Binary collisions counts collisions

Roy Glauber
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Theory vs. Experiment

• Theoretical calculations
• Fundamental Input b is chosen at random
• Npart, Ncoll are given by the model
• Particles are generated
• Experimental measurements
• Fundamental Input Particles are measured
• infer Npart , Ncoll , b
• Not a completely theoretical inference
• Based on some robust assumptions
• Can estimate systematics

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Assumptions

• Monotonicity
• monotonic relationship (on average) between X and
  the measured quantity (N)
• d?X?/d ?N? gt 0 (or lt 0) for all ?N?
• A given fraction of events selected from N should
  correspond to ?X? for the same fraction

Top 6
Top 6
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Centrality with Paddles
e.g., top 6 most central collisions
Entries
3lthlt4.5
Events/Bin
h
Energy deposited in paddle counters
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Proof of Monotonicity
ZDC sum vs. Paddle sum Independent methods to
determine centrality that correlate well
ZDC
ZDC Sum (au)

• See J.M. Katzy in Parallel Session II for details

central
peripheral
Paddle Sum (au)
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Systematic Error

• Total Cross Section
• Fraction of what? (i.e. model has correct sTOT)
• RHIC experiments dont measure sTOT
• Instead trigger efficiency is estimated using
  models (e.g. HIJING)
• Systematic error based on this is included in the
  final results. Large at low Npart!

PHOBOS
3 Uncertainty in sTOT ? 20 Uncertainty in
Npart
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Experimental Problems

• Estimating Npart is not trivial, even if we make
  it seem/sound like it is!
• Ncoll is even harder, since it is not constrained
  by 2A, but by A2!
• Critical to know the total cross section
• But even theorists have trouble with this value!
• Monte Carlo vs. Optical-limit approaches

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Nuclear Profile Thickness
Electron Scattering Measurements
H. DeVries, C.W. De Jager, C. DeVries, 1987
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Total AB Cross Section
Configuration Space
Nuclear Thickness
Interaction Terms
Intractable. Instead, most people use optical
limit
where
Supposedly valid for large A and/or when sNN is
small
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Calculating Npart and Ncoll

• Number of participants
• Number of collisions

This is not the only way to derive these
quantities!
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Glauber Monte Carlo

• Random impact parameter, nucleon positions
• Interactions occur for D lt sqrt (sNN/p)
• Can directly count Npart, Ncoll for each event

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Fun with MC Glauber

• Anyone can program a Monte Carlo Glauber model
  I did!
• Once you have control over where the nucleons
  are, you can study many interesting geometrical
  quantities
• Npart, Ncoll number of interactions
• Area, Eccentricity shape of interaction region
  (figures into v2)
• Fluctuations of these quantities
• While may not perfectly map onto reality, good to
  develop intuitions

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Two Different Answers?

• Kharzeev/Nardi
• Optical-limit approach
• Point nuclei

• HIJING 130 GeV
• Monte Carlo approach
• Gaussian nucleon profile

nucl-ex/0105011
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Beyond the Optical Limit

• Franco Varna, 1977
• Include higher order terms in optical limit
• Next higher order decreases cross section
• Addition of next term increases it again
• Not clear if series is convergent
• Shows difficulty of problem

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Ball Glauber

• Keep optical limit formalism but give nucleons
  finite size
• Modifies nuclear thickness function
• Overlap function
• Total cross section
• Number of participants

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Study of systematics

• We have implemented our own
• Monte Carlo (MC) Glauber calculation
• Numerical Integration
• Point-like nucleons
• Ball-like nucleons
• Allows us to directly compare quantities
• Total cross section
• Npart as a function of percentile of cross
  section
• What are the sensitivities to R, a, sNN

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Total Cross Section
Standard Parameters
Ball optical
Point optical
Monte Carlo
sTOT 6.7 b
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Dilute Limit Convergence

• As we lower the NN cross section, two methods
  converge!
• Reduces effects of multi-particle collisions

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Proton Charge Distribution

• Electron experiments measure charge distribution
• Density r(s) tracks nucleons
• Can do calculation
• Nucleons distributed by 2-parameter Woods-Saxon
• Nucleon charge distributed in gaussian sphere
  (Rp.8fm)
• Recalculate a for original a

• a0 induces a .24 fm
• Adds in quadrature
• a(input) .48 gives a.535
• Gives sTOT 6.5 b

Rp
R
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Impact parameter dependence
Ncoll
Ncoll
Npart
Npart 2
Ncoll 1
Npart
Impact Parameter
Impact Parameter

• Using standard parameters
• Both approaches yield same Npart(b), Ncoll(b)
• We have fixed Npart to prevent Npartlt2, not Ncoll
• Npart(b) x (1-P0(b)) where P0(b) exp(-ABsNNTAB)

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Npart in -ile bins

• Although Npart(b) and Ncoll(b) agree, bins in
  -ile bins disagree systematically
• Clear separation in reasonable ranges in R,s
• Need to lower a substantially to even approach
  agreement
• Total cross section is clearly not the only
  controlling parameter for Npart in bins

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Theoretical Problems!

• Two kinds of Glauber calcuations are done
• Optical Limit Monte Carlo
• Agree on certain features
• Npart(b), Ncoll(b)
• Disagree on others
• Total cross section 6.5 b vs. 7-7.2 b
• Extraction of Npart vs. fraction
• Connection between these is not completely clear
• Two approaches converge in dilute limit (sNN?0)
• Side-point about skin thickness
• a .48 fm might be better than the standard
• Not critical, but could use some clarification

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And it gets worse

• In the Glauber models discussed here there is
  only one cross section
• No mention of other processes
• Elastic shouldnt be important
• Diffractive quasi-elastic, so makes particles
• x-sections not precisely known but are not
  negligible
• If they contribute to the measured cross section,
  can change meaning of sTOT
• Depends on detector configuration
• Combination of theoretical and experimental
  uncertainties
• Only realized recently

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Event Selection
Paddle Counters
Coincidence (38 ns) between paddle counters
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Conclusions

• Experimental Theoretical Problems
• although both types hurt experimentalists!
• Npart and Ncoll are resonably robust for central
  events
• Potentially large uncertainties for peripheral
  events
• Can depend on the input Glauber model and what
  interactions are included
• An important foundation of the field
• Crucial to understand how it works!