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

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

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) 044902, nucl-th/0410036 M. Chojnacki, W.Florkowski, PRC 74 (2006) 034905, nucl-th/0603065 M. Chojnacki, W.Florkowski, Acta. Phys. Polon. B38 (2007) 3249-3262, nucl-th/0702030

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

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

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

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/0801.4361; nucl-th/0710.5731

Full centrality dependence Filled points: STAR data from: Phys. Rev. Lett. 93 (2004) 012301 e-Print Archives (nucl-ex/0312009) 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.

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 0 2 + 𝑇 𝑚 𝑡 r0 – Gaussian system size β0 – maximum flow velocity (linear profile)

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: 0909.5349

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

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

Asymmetry from time difference pT (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:0808.3363 [nucl-th] arXiV: 0909.5349

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

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 ∗ 2 1+cos θ ∗ 2 𝑎 +…

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

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)

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

Sensitivity to asymmetry The Re{C11} component is monotonically dependent on emission asymmetry – a very clean signal Other components not affected, C00 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

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: 0909.5349

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

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: 0909.5349

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: 0909.5349

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)

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.