# Presentation: 004111

## A Tale of Two Glauber Calculations: Monte Carlo vs. Optical Limit Approaches

**Mark Baker, **_**Peter Steinberg**_
- Brookhaven National Laboratory
**Patrick Decowski**
- Massachusetts Institute of Technology

## Theory vs. Experiment

- Theoretical calculations
- _Fundamental Input:_*** ******b***** is chosen at random**
- Npart, Ncoll are derived
- Particles are generated
- Experimental measurements
- _Fundamental Input:_** ****Particles are measured**
- derive Npart , Ncoll ,  b
- Not a completely theoretical derivation
- Based on some robust assumptions
- Can estimate systematics

## Assumptions

- Monotonicity
_***monotonic***_ relationship (on average) between X and the measured quantity (N)
*d*X/*d *N > 0 (or < 0) for all N
- A given fraction of events selected from N corresponds to X for the same fraction
- Total Cross Section
- Fraction of *what*? (i.e. model has correct TOT)
- RHIC experiments have not used the directly measured cross section (see A. Drees talk)
- Instead trigger efficiency is estimated using models (e.g. HIJING)
- Systematic error based on this is included in the final results

## 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 measured in pp collisions, even after interacting
- Participants – counts nucleons which interact
- Binary collisions – counts collisions
- Two methodologies satisfy basic criteria
- Probability integrals
- Monte Carlo approach
- Do they give the same answer?

- Roy Glauber

## Nuclear Profile & Thickness

- H. DeVries, C.W. De Jager, C. DeVries, 1987

## 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 NN is small

## Calculating Npart and Ncoll

- Number of participants
- Number of collisions

## Glauber Monte Carlo

- Random impact parameter, nucleon positions
- Interactions occur for *D < sqrt** **(****NN****)*
- Can directly count Npart, Ncoll for each event

## Two Different Answers?

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

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

- nucl-ex/0105011

## 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

## “Ball” Glauber

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

## 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, NN

## Total Cross Section

- Point optical

- Ball optical

- Monte Carlo

**Standard Parameters:**

******TOT**** = 6.7 b**

## Dilute Limit: Convergence

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

## Proton Charge Distribution

**Proton Charge Distribution**

- Electron experiments measure *charge* distribution
- Density (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

- a~0 induces a ~ .24  fm
- Adds in quadrature
- a(input) = .48 gives a~.535
- Gives ******TOT**** = 6.5 b**

- R

- Rp

- (input)

- (output)

## Impact parameter dependence

- Using standard parameters
- Both approaches yield same Npart(b), Ncoll(b)
- We have fixed Npart to prevent Npart<2, not Ncoll
- Npart(b) x (1-P0(b)) where P0(b) = exp(-ABNNTAB)

- Ncoll

- Npart

- Npart

- Ncoll

- Ncoll = 1

- Npart = 2

- Impact Parameter

- Impact Parameter

## Npart in %-ile bins

**Npart in %-ile bins**

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

## Conclusions

- 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 (NN0)
- Side-point about skin thickness
- a ~ .48 fm might be better than the standard
- Not critical, but could use some clarification