1. Femtoscopic signatures of collective behavior as a probe of the thermal nature of relativistic heavy ion collisions
Thomas J. Humanic,
Ohio State University
Adam Kisiel,
CERN

2. Outline
Motivation: testing the necessity for strong assumptions of hydrodynamics
Simple rescattering model
Quantifying collectivity
Probing collectivity via femtoscopy
Emission asymmetries for non-identical particles
Realistic rescattering model
Cross-checks of the simple model
Realistic simulation for RHIC conditions

3. Does femotscopy probe collectivity?
Hydrodynamics is effective in describing spectra and elliptic flow, what about femtoscopy?
Requires strong assumptions (zero mean free path, local thermalization, “large” system)
Relaxing assumptions seems not to affect femtoscopy – is it really probing the collectivity (and hence thermalization) of the system?
Gombeaud C., Lappi T., Ollitrault J., Phys.Rev.C79:054914, arXiv:  [nucl-th]

4. Simple rescattering model
A. Kisiel, T.J. Humanic; arXiv:
Event consists of 1000 “pions” and 100 “kaons”, initial distribution of: , temperature distribution: and Hubble-like flow in z direction:
Two temperature scenarios: “uniform” - Tmax = Tbase = 300 MeV and “gradient” Tmax = 500 MeV, Tbase= 100 MeV
𝑑𝑁 𝑑𝑥𝑑𝑦𝑑𝑧 ≈exp − 𝑥 2 + 𝑦 2 2R 2 Θ 𝑧 𝑚𝑎𝑥 −𝑧 Θ 𝑧 𝑚𝑎𝑥 +𝑧
𝑇 𝑥,𝑦 = 𝑇 𝑚𝑎𝑥 − 𝑇 𝑏𝑎𝑠𝑒 exp − 𝑥 2 + 𝑦 2 2R 𝑇 𝑇 𝑏𝑎𝑠𝑒
𝑣 𝑧 =𝑎∗𝑧+𝑏

5. Rescattering simulation
Controlling parameter: interaction distance d. We perform calculations for d=0.1, 1.0, 2.0, 5.0, 10.0 fm
For each time slice (0.1 fm/c) we check all pairs. Scattering occurs if the distance of closest approach is less than d, is within time slice, and
Scatterings are elastic and relativistic. Scatterings in the same time slice are carried out simultaneously. Procedure continues until no scattering remain.
Position of last scattering is the “emission point” which is saved, together with final momentum.
𝑟 1 − 𝑟 𝑣 1 − 𝑣 2 <0
d [fm]
0.1
1.0
2.0
5.0
10.0
<Nc>
0.15
2.2
4.7
9.2
12.9
Initial K=λ/R
~2.2
~1.7
~0.95
~0.46
~0.22

6. Quantifying collectivity
Chojnacki M., Florkowski W.
nucl-th/ , Phys. Rev. C74: (2006)
Hydrodynamics produces collective flow: common velocity of all particles
We test if rescattering alone can produce similar effects
Small cross-section gives no common velocity, large cross- section resembles hydro
𝑣 𝑜𝑢𝑡 = 𝑣 𝑇 𝑟 𝑇 | 𝑟 𝑇 |
𝑣 𝑠𝑖𝑑𝑒 = 𝑣 𝑇 × 𝑟 𝑇 | 𝑟 𝑇 |

7. Single particle spectra
We see the spectra in the “gradient” case
No rescattering produces spectra with non-thermal shapes
Increasing d results in evolution towards a known picture: thermal spectra modified by collective velocity
Pions: concave
Kaons: curving downwards
Closed: pions
Open: kaons d
0.1 fm
1.0 fm
2.0 fm
5.0 fm
10.0 fm

8. Source shape – “gradient” no interactions
We inspect emission patterns vs. velocity direction
For the gradient case expected effect is seen: high pT particles are emitted from a smaller region (a “hot” core). But no apparent shift of average emission position is seen, also as expected.
𝑥 𝑠𝑖𝑑𝑒 = 𝑟 × 𝑣 𝑇 | 𝑣 𝑇 |
𝑥 𝑜𝑢𝑡 = 𝑟 𝑣 𝑇 | 𝑣 𝑇 |
pions
faster
pions
same velocity
kaons

9. Source shape with interactions
When d is increased, one still sees the size decrease with particles pT. But is it still due to temperature gradients or is it flow?
An additional effect is seen: Average emission points of particles are shifted to the positive “out” direction

10. mT scaling = collectivity?
pions
kaons
Therminator prediction Phys.Rev.C73:064902, nucl-th/
“Gradient” case with no interactions shows mT dependence – scenario alternative to collectivity?
Simulations with interactions also show mT dependence, but now kaons follow the same trend as pions
open: uniform
closed: gradient
kaons
kaons
pions
Two competing scenarios to explain the “mT scaling”: temperature gradients with little interactions or collectivity from many interactions

11. Isotropization via interactions
As d increases – the differences between “gradient” and “unfiorm” radii disapear: interactions make the system more isotropic
With many interactions the mT scaling of radii is coming only from collectivity, original temperature gradients are forgotten

12. Asymmetries from collectivity
Emission asymmetry between pions and kaons increases linearly with number of collisions per particle – clean and unambigous signature of collectivity
Initial temperature gradients do not matter – asymmetry depends only on collectivity
μ 𝑜𝑢𝑡 π𝐾 = 𝑥 𝑜𝑢𝑡 π − 𝑥 𝑜𝑢𝑡 𝐾

13. Lessons from a simple model
Interactions produce collective effects. Even a purely microscopic rescattering calculation gives “hydro-like” features for reasonable values of “interactions per particle”
Initial temperature gradients may explain some of the “mT dependence” but fail to explain scaling for all masses and do not produce asymmetries
Emission asymmetries a clean probe of collectivity
No collectivity = no assymmetry
Asymmetry depends linearly on Nc
Insensitive to initial temperature gradients

14. Superposition-rescattering model for A+A collision
Initial p+p “thin disk”
of radius r = 1 fm
i-th particle
j-th pp
collision A A
* Superimpose f*A PYTHIA pp events, where f=overlap fraction for impact param.
* Assume all particles have the same proper time for hadronization, , so that
the hadronization space-time for each particle is given by “geometry+causality”, i.e.
ti = Ei/mi ; xi = xoi +  pxi/mi ; yi = yoi +  pyi/mi ; zi =  pzi/mi
* Perform hadronic rescattering using a “full” Monte Carlo rescattering calculation

15. Time evolution of hadronic rescattering
Rescattering hadron
Frozen-out hadron
t ~ 1 fm/c
All hadrons are
rescattering
t ~ 10 fm/c
Some hadrons have
stopped rescattering,
i.e. “frozen out”
t ~ 30 fm/c
Most hadrons have stopped
rescattering --> “freezeout”

16. Other details of the model….
Use PYTHIA v.6409 to generate hadrons for 200 GeV (and 5.5 TeV) p+p “minimum bias” (non-elastic and non-diffractive) events
“final” hadrons from PYTHIA to use:  K, N, , K* ’
Use mult > 20 cut on p+p events --> cuts out ~ 20-25% of events and helps dn/dagree better with PHOBOS 200 GeV/n Au+Au
Monte Carlo hadronic rescattering calculation:
Let hadrons undergo strong binary collisions (elastic and inelastic) until the system gets so dilute that all collisions cease. Use 0.5 fm/c timesteps to 200 (400) fm/c.
--> isospin-averaged (i,j) from Prakash, etc..
Record the time, mass, position, and momentum of each hadron when it no longer scatters.  freezout condition
Take  = 0.1 fm/c for all calculations --> found to agree with Tevatron HBT data (T. Humanic, Phys.Rev.C76, (2007))
Au+Au Model calculations found to describe observables from RHIC experiments reasonably well (T. Humanic., Phys.Rev.C79, (2009)).

17. for minimum bias sqrt(sNN)=200 GeVAu+Au collisions.
Time evolution up to 50 fm/c of the particle density calculated at mid-rapidity
( −1 < y < 1) and the number of rescatterings per time step from the model
for minimum bias sqrt(sNN)=200 GeVAu+Au collisions. {
Big assumption:
binary collisions
of hadrons
or hadron-like
particles
here, too!!
200
fm/c

18. Spectra: Model vs PHOBOS, PHENIX, and STAR
Model describes
the trends of the
spectra data
reasonably well.
Absolute normalizations used

19. v2/nq vs. pT/nq for Model vs. PHENIX For pions, kaons, and protons.
Model v2 shows
scaling with quark
number like exp.

20. Azimuthal  HBT: Model vs. STAR
Model describes and multiplicity dependences of
HBT reasonably well.

21. Repeating simple model calculations
The simple model calculations have been repeated with the more realistic rescattering
The “uniform” and “gradient” initial conditions have been used as a cross-check with realistic rescatterings (both elastic and inelastic)
In addition a “realistic” scenario described before was used as well, to test if the purely hadronic model will give the same signatures

22. Calculating system size
All particles are grouped in pairs and the separation distribution is calculated
Fitting with a gaussian around the peak of the rescattering model dN/Δxout distribution for pions to extract the xout widths for two mTave ranges.
Similar procedure for the asymmetries

23. Size trends for realistic model
Both gradient and “pp superposition” produce mT dependence, uniform does not
With realistic cross- sections kaons sizes are close to pions (universal) Differences biggest for “gradient” scenario

24. Asymmetry in realistic model
Realistic cross-section gives large asymmetry
Uniform and gradient case give different asymmetries in contrast to the simple model
Superposition model gives largest asymmetry, comparable to the uniform

25. Comparing emission points
Difference between gradient and uniform: much larger pion shift:
With higher temperature larger rate of inelastic rescattering via rho – larger shift for pions
Uniform case more similar to the superposition
The “realistic” simulation favors the uniform case

26. Summary
Simple rescattering model: complementary approach to studies on relaxing hydrodynamic assumptions
Collective “hydro-like” effects develop with sufficient amount of interactions
Dependence of femtoscopic radii on mT not unambiguous signature of collectivity. Must utilize mass dependence: “universal” scaling for pions and kaons
Emission asymmetries most sensitive to rescatterings: unambiguous probe of collectivity
“Realistic” rescattering model gives similar results, favors the “unfiorm temperature” initial condition