1. Non-identical particle femtoscopy in hydrodynamics with statistical hadronization
Adam Kisiel,
CERN

2. Outline
Hydrodynamics with statistical hadronization – Lhyquid+Therminator model
Why measure emission asymmetries: direct femtoscopic evidence for collective matter behavior
Non-identical particle formalism
Femtoscopy via Coulomb interaction
How to measure emission asymmetries
Simulation results
Asymmetries for all pair types and vs. centrality
Internal consistency and data cross-checks

3. Lhyquid: 2+1 hydro 2+1 hydrodynamics code with boost-invariance
Smooth equation of state formed by merging hadronic gas and lattice calculation parts
Smooth cross-over – no sharp phase transition: leads to faster evolution times and smaller sources at T<Tc
Initial conditions can be asymmetric – naturally producing elliptic flow and azimuthally sensitive femtoscopy
Temperature at the beginning of the evolution Ti and freeze-out temperature Tf are free parameters
M. Chojnacki, W.Florkowski, T.Csorgo, PRC 71 (2005) , nucl-th/
M. Chojnacki, W.Florkowski, PRC 74 (2006) , nucl-th/
M. Chojnacki, W.Florkowski, Acta. Phys. Polon. B38 (2007) , nucl-th/

4. Lhyquid: initial conditions
Initial transverse entropy density from Glauber as a sum of wounded nucleon and binary collision densities
Assumed temperature profile is
Initial time τinit= 0.25 fm.
Another option: width taken from Glauber but use Gaussian shape.
𝑠 𝑥 𝑡 =ρ 𝑥 𝑡 = 1−𝐴 2 ρ 𝑊 𝑥 𝑡 +𝐴 ρ 𝑏𝑖𝑛 𝑥 𝑡
𝑇 τ 𝑖𝑛𝑖𝑡 , 𝑥 𝑡 = 𝑇 𝑆 𝑠 𝑖 ρ 𝑥 𝑇 ρ 0
𝑛 𝑥,𝑦 =exp − 𝑥 2 2 𝑎 2 − 𝑦 2 2 𝑏 2

5. Freeze-out hypersurface
Freeze-out occurs at an isotherm T=145 MeV, the shape is determined by hydrodynamic calculation
Non-azimuthally symmetric shapes arise, as well as substantial radial flow. Initial shape matters.
fluid velocity at
freeze-out
RHIC
RHIC
LHC

6. Freeze-out and treatment of resonances
At freeze-out system is converted to particles via the von Neumann sampling of the Cooper-Frye formula
Per-event particle multiplicities randomly generated with averages taken from the chemical model
Resonance propagation and decay included after freeze- out, but not rescattering
All resonances from the PDG (381 types) included
Propagation and decay done particle by particle, in cascades
All pion sources
compared

7. Lhyquid+Therminator at RHIC
Dynamical model with hydrodynamical evolution and strong resonance propagation
Reproduces spectra, elliptic flow and HBT
Includes all effects important for proper non- id asymmetry predictions
W.Broniowski, W.Florkowski, M.Chojnacki, AK
nucl-th/ ; nucl-th/

8. Full centrality dependence
Filled points: STAR data from:
Phys. Rev. Lett. 93 (2004)
e-Print Archives (nucl-ex/ )
Colors: different kt bins
Open points: Lhyquid+Therminator
Phys.Rev.C79:014902,2009.
Phys.Rev.C78:014905,2008.
Full centrality vs. pair transverse momentum vs. reaction plane orientation dependence of HBT radii is reproduced.
Model shows very good agreement for a wide range of soft sector observables at RHIC.

9. Thermal emission from flowing medium
A particle emitted from a flowing medium will have a velocity βf from the flow field and a thermal one βt
βf is always pointing “outwards” (space-momentum correlation: φf=φr), βt is random
Average “outwards” emitting point will be then
𝑥 𝑜𝑢𝑡 = 𝑥 β β = 𝑟 0 β 0 β β 𝑇 𝑚 𝑡
r0 – Gaussian system size
β0 – maximum flow velocity (linear profile)

10. Collectivity and emission asymmetry
β = (0.6,0.8)
Pions
As particle mass (or pT) grows, two effects emerge:
System size shrinks (well known: “lenghts of homogeneity”)
Average emission point moves more “outwards”
Average emission points for particles with same velocity but different mass
Randomizing φr (destroying x-p correlation) removes asymmetry
Moving “upwards”
Randomized φr
Kaons
Pions <xπout>
Kaons <xKout>
Protons <xpout>
2.83 fm
4.47 fm
5.61 fm
Protons
Asymmetry:
𝑟 𝑜𝑢𝑡 π𝐾 ≈ 𝑥 𝑜𝑢𝑡 π − 𝑥 𝑜𝑢𝑡 𝐾
arXiV:

11. Resonance decay Resonance decay:
Primordial
Resonance decay
Resonance decay:
Fixed “parent” velocity (coming from flow)
Random decay velocity
Decay momentum matters more for light particles
If pdecay>mdaughter than randomization from decay is strong (loss of x-p correlation), if not: the correlation is preserved
βdecay=pdecay/E
From resonances βr
βtotal
resonance
All

12. Resonances and asymmetry
Primordial
Very different scenarios for pions than for kaons (and protons) from resonances
Pions almost completely randomized
Kaons shifted more than primordial due to parent resonance propagation
Detailed quantitative simulation shows that resonance decay enhances pion-kaon(proton) flow asymmetry
Pions
Kaons
From resonances
All

13. Asymmetry from time differencepT
(0.15,0.25)
(0.25,0.35)
(0.35,0.45)
(0.45,0.6)
High pT particles emitted earlier
Additional time shift from boost- invariance assumption
But also non-hydrodynamic:
More pions from resonances – larger average emission time
Phys.Rev.C79:014902,2009.
arXiv: [nucl-th]
arXiV:

14. Asymmetry: from collectivity or not?
pair
π-K
π-p
K-p
space asymmetry with flow
primordial
-1.64
-2.78
-1.14
all
-3.54
-4.69
-1.15
x-p correlations switched off
0.0
-0.72
-0.80
-0.08
Collective sources:
Flow
Hydrodynamic emission time ordering with pT
Boost-invariance
Non-collective sources
Resonance propagation (<20%)
Resonance time delay (<15%)
At least 2/3 of effect collective
asymmetry
space
time
all
π-K all
-5.0
-3.0
-8.0
primordial
-2.4
-1.7
-4.1
π-p all
-5.7
-3.5
-9.2
-3.4
-2.0
-5.4
K-p all
-1.1
-0.6
-1.3
-0.4

15. Non-identical particle correlations via the coulomb wave-function
|Ψ(q,r)|2 r1 r2
q=p1-p2
𝐶 𝑞 = 𝑆 𝐫 |Ψ 𝑞 ,𝐫 | 2 𝑑 4 𝑟
𝑆 𝐫 = 𝑆 𝑎 𝐫 𝟏 𝑆 𝑏 𝐫 𝟐 δ 𝐫− 𝐫 𝟏 − 𝐫 𝟐 𝑑 4 𝑟 1 𝑑 4 𝑟 2 p1 p2
Ψ − 𝐤 ∗ 𝐫 ∗ = 𝑒 𝑖 δ 𝑐 𝐴 𝑐 η 𝑒 −𝑖 𝐤 ∗ 𝐫 ∗ 𝐹 −𝑖η,1,𝑖ξ + 𝑓 𝑐 𝑘 ∗ 𝐺 ρ,η 𝑟 ∗
Gamow factor
Coulomb part
Strong Interaction part
ξ= 𝐤 ∗ 𝐫 ∗ + 𝑘 ∗ 𝑟 ∗ ≡ρ 1+cos θ ∗ ,ρ= 𝑘 ∗ 𝑟 ∗ ,η= 𝑘 ∗ 𝑎 −1 ,𝑎= μ 𝑧 1 𝑧 2 𝑒 2 −1
Here we use Coulomb only. Femtoscopic sensitivity is in F:
It depends on “size” r* and on the angle θ* vs. relative momentum k*
𝐹 𝑘 ∗ , 𝑟 ∗ , θ ∗ =1+ 𝑟 ∗ 1+cos θ ∗ 𝑎 + 𝑟 ∗ 1+cos θ ∗ 𝑎 2 +𝑖 𝑘 ∗ 𝑟 ∗ cos θ ∗ 2 𝑎 +…

16. Accessing asymmetry Ψ=φ+θ* We want to measure
But we only measure relative k* and total momentum v, so we only know Ψ
We also know that the CF depends on θ*
The three angles are connected by a simple sum rule: average cosine signs must also follow
By looking at the CF vs we are able to access asymmetries
𝑘 𝑜𝑢𝑡 ∗ ≡ 𝑘 ∗ cos Ψ 𝑘 𝑠𝑖𝑑𝑒 ∗ ≡ 𝑘 ∗ sin Ψ
Transverse plane
𝑟 𝑜𝑢𝑡 ∗ ≡ 𝑟 ∗ cos φ
2k* p2 p1 v Ψ x2 x1 r*
2k* p2 p1 v θ* φ
Ψ=φ+θ*
cos Ψ =cos ϕ cos θ ∗ +sin ϕ sin θ ∗
cos Ψ
𝑠𝑖𝑔𝑛 cos Ψ =𝑠𝑖𝑔𝑛 cos ϕ 𝑠𝑖𝑔𝑛 cos θ ∗
𝑖𝑓|𝐶 cos Ψ >0 −1|>|𝐶 cos Ψ <0 −1|𝑡ℎ𝑒𝑛 cos φ <0

17. CF (in Spherical Harmonics)
𝐶 𝑙 𝑚 𝑞 = 𝐶 𝑞,cos θ 𝑞 , φ 𝑞 𝑌 𝑙 𝑚 cos θ 𝑞 , φ 𝑞
All imaginary vanish
For rapidity symmetric collider systems: all odd cos(θ) vanish
C00 – angle averaged
C11 – asymmetry
C20 – Rlong/Rtransverse
C22 – Rout/Rside
All others can be neglected
correlation function in spherical harmonics
Y00~1
Y10~cos(θq)
Y11~sin(θq)cos(φq)
Y20~cos2(θq)
Y22~sin2(φq)

18. Sensitivity to size
The size of the system is mostly reflected in the C00 component (“overall strength of correlation”)
Residual sensitivity in Re{C11} as well
C00 is the correlation function averaged over a fixed q sphere
A good monotonic dependence enables size extraction, but only for “not too large” sizes
Ro=8 fm
Rs=4 fm
Rl=4 fm
μ = -4 fm
Ro=10 fm
Rs=5 fm
Rl=5 fm
μ = -5 fm
Ro=12 fm
Rs=6 fm
Rl=6 fm
μ = -6 fm

19. Sensitivity to asymmetry
The Re{C11} component is monotonically dependent on emission asymmetry – a very clean signal
Other components not affected, C slightly affected
Re{C11} is: cos(φq) weighting maximizes statistical significance
Ro=8 fm
Rs=4 fm
Rl=4 fm
μ = 0 fm
μ = -2 fm
μ = -4 fm
μ = -6 fm
μ = -8 fm
R{ 𝐶 1 1 =𝑁 𝐶 sin θ 𝑞 cos φ 𝑞

20. Characterizing model input
The source has a clear shift in “out”. It is not gaussian.
Can be approximateed with gaussian – sigma is the “model radius”, <r*out> is asymmetry
Resonance decays produce long tails in emission function: the pairs in the tails should be treated as “non-correlated”, they are outside femtoscopically sensitive range of r*
Gaussian fit
% of femtoscopically non-correlated pairs
Centrality
0-5
5-10
10-20
20-30
30-40
40-50 πK 83 82 80 76 73 70 πp 79 77 75 71 Kp 96 95 94 93
arXiV:

21. Fitting non-id CF
Fitting is numerical. A source is assumed to have size σ and shift μ. Theoretical functions are calculated for several sets of (σ,μ). The one with best χ2 is taken as the result of the fit.
π-K
points – CF
from model
π-p
lines - fit
K-p

22. Recovering model input
We use the model to simulate “experimental-like” CF
“Input”: the source size from the model itself
Result of the test: fitting recovers model “input” values.
Crucial systematic effect: correct estimation of the “fraction of non-correlated pairs” a.k.a. purity
arXiV:

23. Predictions from Therminator+Lhyquid
STAR 130 AGeV
Expected centrality trend: size grows, asymmetry grows
Expected size and asymmetry ordering for three pair types
Sizeable asymmetry seen for pion-kaon and pion-proton pairs
pion-kaon
pion-proton
kaon-proton
arXiV:

24. Consistency checks
pion size from HBT (STAR)
If one measures all two-particle source sizes and asymmetries, one can calculate the single particle source sizes – these are compared to RHIC HBT sizes
Asymmetries between all pair types should add up to zero
All possible cross-checks show that model is self-consistent and consistent with RHIC results
kaon size from HBT (PHENIX)
pion, kaon, proton
size from non-id
proton size from HBT (STAR)

25. Summary
Therminator+Lhyquid model used for asymmetry modeling – includes all important components
Non-identical particle correlations provide crucial cross- check of the consistency of system evolution, through the unique feature of measuring emission asymmetries
All contributions to the emission asymmetry studied, at least 2/3 come from collectivity. Estimate for non-collective asymmetry from resonances provided.
Technical cross-checks of the method done.
Shown results for full centrality and pair type dependence at RHIC.