SLIDE 1
Hydrodynamics and femtoscopy in heavy-ion physics
B´ alint Kurgyis
E¨
- tv¨
- s University, Budapest
Hydrodynamics and femtoscopy in heavy-ion physics B alint Kurgyis - - PowerPoint PPT Presentation
Hydrodynamics and femtoscopy in heavy-ion physics B alint Kurgyis - - PowerPoint PPT Presentation
Hydrodynamics and femtoscopy in heavy-ion physics B alint Kurgyis E otv os University, Budapest Bolyai Physics Seminar Budapest, 20 February 2019 Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Big
SLIDE 2
SLIDE 3
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary
Big bang in the laboratory - heavy-ion collisions
Timeline of the Universe Galaxies Atoms Nuclei Elementary particles How should we investigate? Reproduce in the laboratory! Create “little bangs” → Heavy-ion collisions Detect the created particles → Study the sQGP
B´ alint Kurgyis Hydrodynamics and femtoscopy 3 / 38
SLIDE 4
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary
The strongly interacting quark-gluon plasma (sQGP)
Discovered at the RHIC, created at LHC Hot, expanding, perfect quark-fluid Hadrons created at the freeze-out Photons and leptons ”shine through”
B´ alint Kurgyis Hydrodynamics and femtoscopy 4 / 38
SLIDE 5
SLIDE 6
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
Equations of non-relativistic hydrodynamics
Looking for (u, p, ρ) fields Assumptions: zero viscosity zero heat conductivity Euler-equation
∂u ∂t + (u∇)u = − 1 ρ∇p
Continuity equation
∂ρ ∂t + ∇(ρu) = 0
Equation of state p − ρ relation
B´ alint Kurgyis Hydrodynamics and femtoscopy 6 / 38
SLIDE 7
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
Perturbative equations
Perturbed fields u → u + δu p → p + δp ρ → ρ + δρ Perturbed equations first order perturbation using another solution
B´ alint Kurgyis Hydrodynamics and femtoscopy 7 / 38
SLIDE 8
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
Wave solution
Known solution: Standing fluid u = 0 p = const. ρ = const. Sound speed from equation of state: ∂p ∂ρ = c2 Perturbed Euler-equation
∂δu ∂t = − 1 ρ∇δp
Perturbed continuity equation
∂δρ ∂t + ρ∇δu = 0
Wave solution for pressure
∂2δp ∂t2 = c2∆δp
B´ alint Kurgyis Hydrodynamics and femtoscopy 8 / 38
SLIDE 9
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
Equations for relativistic hydrodynamics
Looking for (uµ, p, ǫ, n or σ) fields Assumptions: zero viscosity zero heat-conductivity local energy-momentum conservation Properties: uµuµ = 1 gµν = diag(1, −1, −1, −1) Temperature: T = (ǫ + p)/σ Locally conserved entropy density n → conserved charge density Energy, momentum conservation ∂µT µν = 0 T µν = (ǫ + p)uµuν − pgµν Continuity equation ∂µ(nuµ) = 0 or ∂µ(σuµ) = 0 Equation of state (EoS) ǫ = κp
B´ alint Kurgyis Hydrodynamics and femtoscopy 9 / 38
SLIDE 10
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
Known solutions for relativistic hydrodynamics
Many numerical solutions Exact, analytic solutions important: connect initial/final state Famous 1+1D solutions: Landau-Khalatnikov & Hwa-Bjorken
- L. D. Landau, Izv. Akad. Nauk Ser. Fiz. 17, 51 (1953)
I.M. Khalatnikov, Zhur. Eksp. Teor. Fiz. 27, 529 (1954)
- R. C. Hwa, Phys. Rev. D 10, 2260 (1974)
- J. D. Bjorken, Phys. Rev. D 27, 140 (1983)
Discovery of sQGP → Many new solutions First truly 3D relativistic solution: Hubble-flow
Cs¨
- rg˝
- , Csernai, Hama, Kodama, Heavy Ion Phys. A21, 73 (2004), nucl-th/0306004
Describes well experimental data
Csan´ ad, Vargyas, Eur. Phys. J. A 44, 473 (2010) nucl-th/09094842
B´ alint Kurgyis Hydrodynamics and femtoscopy 10 / 38
SLIDE 11
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
Perturbative handling of the relativistic hydrodynamics
Perturbed fields: Start from a known solution: (uµ, p, n) uµ → uµ + δuµ p → p + δp n → n + δn works similarly for n → σ Orthogonality: uµδuµ = 0 (1) Equations for perturbations: substitute perturbations into equations substract 0th order equations neglect 2nd or higher order perturbations remainder: perturbed equation solution yields perturbations δuµ, δn, δp
B´ alint Kurgyis Hydrodynamics and femtoscopy 11 / 38
SLIDE 12
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
Known solution: Hubble-flow
Hubble-flow:
Cs¨
- rg˝
- , Csernai, Hama, Kodama , Heavy Ion Phys. A21, 73 (2004), nucl-th/0306004
uµ = xµ
τ
n = n0 τ0
τ
3 N(S) p = p0 τ0
τ
3+ 3
κ
Scaling variable: uµ∂µS = ∂τS = 0 Describes well hadronic data and photons
Csan´ ad, Vargyas, Eur. Phys. J. A 44, 473 (2010) nucl-th/09094842 Csan´ ad, M´ ajer, Central Eur. J. Phys. 10 (2012)
Multipole solutions also possible
Csan´ ad, Szab´
- , Phys. Rev. C 90, 054911 (2014)
B´ alint Kurgyis Hydrodynamics and femtoscopy 12 / 38
SLIDE 13
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
Finding solutions for perturbations
The way of solution: Choosing test functions: δp, δuµ, δn Fixing arbitrary functions Choosing scaling variable S Satisfying all the restrictions instead, one could look for sound waves on top of Hubble-flow
- S. Shi, J. Liao, P. Zhuang Phys.Rev. C90 no.6, 064912 (2014) arXiv:1405.4546
Perturbations: uµ = xµ τ →δuµ = δ · F(τ)g(xν)∂µS · χ(S) (2) n = n0 τ0 τ 3 N(S) →δn = δ · n0 τ0 τ 3 h(xν)ν(S) (3) p = p0 τ0 τ 3+ 3
κ
→δp = δ · p0π(S) τ0 τ 3+ 3
κ
(4)
B´ alint Kurgyis Hydrodynamics and femtoscopy 13 / 38
SLIDE 14
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
General form of solutions
Perturbations: δuµ = δ · F(τ)g(xν)∂µS · χ(S) (5) δp = δ · p0π(S) τ0 τ 3+ 3
κ
(6) δn = δ · n0 τ0 τ 3 h(xν)ν(S) (7)
δuµ → defined by g(xν) χ(S) and F(τ) fixed by g(xν) δp → π(S) determined by δuµ δn → ν(S) fixed by h(xν) g(xν), h(xν), S need to fullfill (8)-(10)
Equations for N(S), χ(S), ν(S), π(S), h(xµ), g(xµ), S functions χ′(S) χ(S) = − ∂µ∂µS ∂µS∂µS − ∂µS∂µ ln g(xν) ∂µS∂µS (8) π′(S) χ(S) = (κ + 1)
- F(τ)
- uµ∂µg(xν) − 3g(xν)
κτ
- + F ′(τ)g(xν)
- (9)
ν(S) χ(S)N ′(S) = −F(τ)g(xν)∂µS∂µS uµ∂µh(xν) (10)
B´ alint Kurgyis Hydrodynamics and femtoscopy 14 / 38
SLIDE 15
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
Looking for concrete solution
To compute measurables → fix the g(xν), F(τ), h(xν) functions: g(xν) = 1, F(τ) = τ + cτ0 τ τ0 3
κ
h(xν) = ln
- τ
τ0
- + c
κ 3−κ
- τ
τ0
3
κ −1
if κ = 3 (1 + c) ln
- τ
τ0
- if κ = 3
Restrictions: uµ∂µS = 0
∂µ∂µS ∂µS∂µS is a function of the scaling variable
τ 2∂µS∂µS also a function of the scaling variable Scaling variables found so far: S = rm
tm ,
S = rm
τ m ,
S = τ m
tm
B´ alint Kurgyis Hydrodynamics and femtoscopy 15 / 38
SLIDE 16
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
Scaling variable: S = r m/tm
Scaling variable S = rm tm (11) The functions of the scaling variable: Perturbations
δp = δ · p0 τ0
τ
3+ 3
κ π(S)
δuµ = δ ·
- τ + cτ0
- τ
τ0
3
κ
∂µSχ(S) δn = δ · n0 τ0
τ
3
- ln
- τ
τ0
- + c
κ 3−κ
- τ
τ0
3
κ −1
ν(S)
χ(S) = r t −m−1 (12) π(S) = −(κ + 1)(κ − 3) κ m r t −1 (13) ν(S) = m2 r t m−1 r t 2 − 1 1 − r t −2 N ′ rm tm
- (14)
B´ alint Kurgyis Hydrodynamics and femtoscopy 16 / 38
SLIDE 17
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
Concrete solution with S = t/r, N(S) = exp(−S−2)
Particular case: m = −1 S = t r (15)
Let us choose Gaussian N(S): N(S) = e− r2
t2 = e−S−2
The functions of the scaling variable: χ(S) = 1 (16) π(S) = (κ + 1)(κ − 3) κ t r
- (17)
ν(S) = 2 t r −3 1 − t r 22 N t r
- (18)
B´ alint Kurgyis Hydrodynamics and femtoscopy 17 / 38
SLIDE 18
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
Four-velocity perturbation
δuµ = δ ·
- τ + cτ0
- τ
τ0
3
κ
∂µS uµ = xµ
τ
Used parameters (describes v2, N(pT), RHBT):
- M. Csan´
ad, M. Vargyas, Eur. Phys. J. A 44, 473
0.2 0.4 0.6 0.8 1 1.2 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ux+δux x [fm] δ=0 δ=0.001, c=-3 τ=6 fm/c 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 1 1.5 2 2.5 3 3.5 4 4.5 5 (ux+δux)/ux x [fm] δ=0 δ=0.001, c=-3 δ=0.001, c=2, δ=0.0005, c=-3 δ=0.0005, c=2 τ=6 fm/c
B´ alint Kurgyis Hydrodynamics and femtoscopy 18 / 38
SLIDE 19
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
Pressure perturbation
δp = δ · p0 τ0
τ
3+ 3
κ (κ+1)(κ−3)
κ
S p = p0 τ0
τ
3+ 3
κ 2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 2.5 1 2 3 4 5 6 7 8 p+δp x [fm] δ=0 δ=0.001 τ=6 fm/c 0.8 1 1.2 1.4 1.6 1.8 2 1 2 3 4 5 6 (p+δp)/p x [fm] δ=0 δ=0.01 δ=0.005 δ=0.001 δ=0.0005 τ=6 fm/c
B´ alint Kurgyis Hydrodynamics and femtoscopy 19 / 38
SLIDE 20
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
Particle density perturbation
δn = δ · 2bn0 τ0
τ
3
- ln
- τ
τ0
- + c
κ 3−κ
- τ
τ0
3
κ −1
S−3 1 − S22 N(S) n = n0 τ0
τ
3 N(S)
0.5 1 1.5 2 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 n+δn x [fm] δ=0 δ=0.001, c=-3 τ=6 fm/c 0.4 0.6 0.8 1 1.2 1.4 1.6 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 (n+δn)/n x [fm] δ=0 δ=0.01, c=-3 δ=0.01, c=2 δ=0.005, c=-3 δ=0.005, c=2 τ=6 fm/c
B´ alint Kurgyis Hydrodynamics and femtoscopy 20 / 38
SLIDE 21
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
Single-particle distribution
Source function → J¨ uttner-distribution: S(x, p)d4x = Nn exp
- −pµuµ
T
- H(τ)pµd3Σµ(xµ)dτ
(19) The Cooper–Frye factor:pµd3Σµ(xµ) = pµuµ
u0 d3x
Freeze out at const. proper time (τ0)→ H(τ) = δ(τ − τ0) With the perturbations: S(x, p) = Nn exp
- −pµuµ
T
- δ(τ − τ0)pµuµ
u0 · (1 + ∆)dτdx3 ∆ = δu0 u0 + pµδuµ pνuν − pµδuµ T + pµuµδT T 2 + δn n
- Single-particle distribution:
N1(p) =
- S(x, p)d4x
(20)
B´ alint Kurgyis Hydrodynamics and femtoscopy 21 / 38
SLIDE 22
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
Single-particle transverse momentum distribution, S = t/r
Two component Gaussian: N(p) = Nn0E1V1(1 + P1 + P2 + P3) + Nn0E2V2(P4 + P5) E1 = exp
- − E 2+m2
2ET0 − p2 2ETeff
- E2 = exp
- − E 2+m2
2ET0 − p2 2ETeff,δ
- Used parameters: describes hadronic & photonic data (v2, RHBT, N(pT))
- M. Csan´
ad, M. Vargyas, Eur. Phys. J. A 44, 473 (2010)
0.96 0.98 1 1.02 1.04 100 200 300 (N1(pt)+δN1(pt))/N1(pt) pt [MeV] δ=0, c=0 δ=0.1, c=4 δ=0.5, c=-3 δ=0.5, c=7 δ=0.5, c=-9
Effective temperatures: Teff = T0 + T0E ˙ R0
2
2b(T0 − E) Teff,δ = T0 + T0E ˙ R0
2
2b(2T0 − E)
B´ alint Kurgyis Hydrodynamics and femtoscopy 22 / 38
SLIDE 23
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Perturbative handling The new class of solutions Observables
HBT-radii for S = t/r
Size of the source → HBT-radii R−2 ∝ mt scaling R2
HBT = T0τ 2 0 (Teff − T0)
ETeff R2
HBT,δ = T0τ 2 0 (Teff,δ − T0)
ETeff,δ
2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 200 400 600 800 RHBT, RHBT,δ [fm] pt [MeV/c] RHBT(pt) RHBT,δ(pt) 0.02 0.04 0.06 0.08 0.1 0.12 0.14 200 400 600 800 1/RHBT
2, 1/RHBT,δ 2 [1/fm2]
mt [MeV/c2] 1/RHBT
2(pt)
1/RHBT,δ
2(pt)
Parameters: δ = 0.5, c = −3
B´ alint Kurgyis Hydrodynamics and femtoscopy 23 / 38
SLIDE 24
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Three-dimensional L´ evy HBT Kinematic variables 3D L´ evy HBT at PHENIX
The HBT effect and the Bose-Einstein correlations
- R. Hanbury Brown, R. Q. Twiss - radio telescopes
Intensity correlations as function of detector distance → Measuring the size of the source (Sirius)
Goldhaber et al. - application in high energy physics
Bose-Einstein correlations - momentum correlations Related to the source function C(q) ∼ = 1 + |
- S(r)eiqrdr|2, where q = p2 − p1
Measuring momentum correlations → femtoscopic space-time geometry
B´ alint Kurgyis Hydrodynamics and femtoscopy 24 / 38
SLIDE 25
SLIDE 26
SLIDE 27
SLIDE 28
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Three-dimensional L´ evy HBT Kinematic variables 3D L´ evy HBT at PHENIX
The PHENIX experiment
Different collision energies
7.7-200 GeV in √sNN 20-400 MeV in µB
Different collision systems
p+p, p+A, A+A
This analysis: 200 GeV Au+Au
particle emitting source space-time evolution of sQGP
B´ alint Kurgyis Hydrodynamics and femtoscopy 28 / 38
SLIDE 29
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Three-dimensional L´ evy HBT Kinematic variables 3D L´ evy HBT at PHENIX
L´ evy distribution and the shape of the correlation function
Expanding medium → increasing mean free path
→ Anomalous diffusion
L´ evy-stable distribution (generalized cent. lim. theor.) L(r; α, R) = 1 (2π)3
- d3qeiqre− 1
2 |qR|α
Power-law tail: ∝ r−1−α L´ evy exponent: α (Gaussian: α = 2, Cauchy: α = 1) Two component source:
Cs¨
- rg˝
- , T. and L¨
- rstad, B. and Zim´
anyi, J., Z. Phys. C71 , 491 (1996), hep-ph/9411307
Core: thermalized medium, expanding source Halo: long lived resonances (τ > 10 fm/c)→ experimentally unresolvable
True q → 0 limit: C(q = 0) = 2; Experimentally: C(q → 0) = 1 + λ Correlation strength: λ =
- NCore
NCore+NHalo
2
- Corr. func.: C(q; R, α, λ) = 1 + λe−|qR|α
B´ alint Kurgyis Hydrodynamics and femtoscopy 29 / 38
SLIDE 30
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Three-dimensional L´ evy HBT Kinematic variables 3D L´ evy HBT at PHENIX
The connection of L´ evy-index and the critical point
Looking for critical behavior with critical exponents Critical spatial correlation: ∼ r−(d−2−η) L´ evy source: ∼ r−(1+α) → η ⇐ ⇒ α ?
Cs¨
- rg˝
- et al., AIP Conf. Proc. 828 525532 (2006)
QCD universality class ⇐ ⇒ 3D Ising
Halasz et al., Phys. Rev. D58 096007 (1998) Stephanov et al., Phys. Rev. Let. 81 4816 (1998)
Critical point: η ≤ 0.50 Motivation for precise L´ evy HBT! Finite size, non-equilibrium effects
What does the power-law tail mean?
B´ alint Kurgyis Hydrodynamics and femtoscopy 30 / 38
SLIDE 31
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Three-dimensional L´ evy HBT Kinematic variables 3D L´ evy HBT at PHENIX
Bersch-Pratt coordinates, LCMS frame
Particles with the same mass (pions), with momentum k1 and k2
Average momentum: K = (k1 + k2)/2 Relative momentum: q = (k1 − k2)
Bertsch-Pratt coordinates: q = (qout, qside, qlong)
Out: direction of average transverse momentum Long: beam direction Side: perpendicular to both
Longitudinally comoving system (LCMS): Klong = 0
qLCMS
- ut
= qxKx + qyKy KT , KT =
- K 2
x + K 2 y ,
qLCMS
side
= qxKy − qyKx KT , mT =
- m2 + K 2
T,
qLCMS
long
=
- 4 (k1zE2 − k2zE1)2
(E1 + E2)2 − (k1z + k2z)2 , qLCMS = KT mT qLCMS
- ut
.
B´ alint Kurgyis Hydrodynamics and femtoscopy 31 / 38
SLIDE 32
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Three-dimensional L´ evy HBT Kinematic variables 3D L´ evy HBT at PHENIX
Three-dimensional correlation function
3D correlations of identified same charged pion pairs Measured in 31 different transverse-mass (mT) bins L´ evy-source: C 0
2 = 1 + λ exp(−|q2
- R2
- + q2
s R2 s + q2 l R2 l |α/2)
Example correlation function → 1D projections Charged pions → (approx.) Coulomb correction
- ut
side long B´ alint Kurgyis Hydrodynamics and femtoscopy 32 / 38
SLIDE 33
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Three-dimensional L´ evy HBT Kinematic variables 3D L´ evy HBT at PHENIX
The size of the source
L´ evy-scale parameter vs. mT describes the size of the source Comparison with 1D L´ evy results
- A. Adare et al. (PHENIX), Phys. Rev. C 97, 064911 arXiv:1709.05649
Source is not spherical Hydro scaling: R ∝ 1/√mT
]
2
[GeV/c
T
m 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R [fm] 2 4 6 8 10 12 PHENIX
- ut
) 3D
- π
- π
(
- ut
R ) 3D
+
π
+
π (
- ut
R ) 1D Phys. Rev. C 97, 064911
- π
- π
R ( ) 1D Phys. Rev. C 97, 064911
+
π
+
π R ( ]
2
[GeV/c
T
m 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 side 0-30 % Centrality ) 3D
- π
- π
(
side
R ) 3D
+
π
+
π (
side
R ) 1D Phys. Rev. C 97, 064911
- π
- π
R ( ) 1D Phys. Rev. C 97, 064911
+
π
+
π R ( ]
2
[GeV/c
T
m 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 long = 200 GeV
NN
s Au+Au ) 3D
- π
- π
(
long
R ) 3D
+
π
+
π (
long
R ) 1D Phys. Rev. C 97, 064911
- π
- π
R ( ) 1D Phys. Rev. C 97, 064911
+
π
+
π R (
PH ENIX
preliminary B´ alint Kurgyis Hydrodynamics and femtoscopy 33 / 38
SLIDE 34
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Three-dimensional L´ evy HBT Kinematic variables 3D L´ evy HBT at PHENIX
The strength of the correlation
]
2
[GeV/c
T
m
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
λ
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 3D
- π
- π
3D
+
π
+
π 1D Phys. Rev. C 97, 064911
- π
- π
1D Phys. Rev. C 97, 064911
+
π
+
π PHENIX 0-30% Centrality = 200 GeV
NN
s Au+Au PH ENIX
preliminary
Correlation strength vs. mT Ratio of resonance pions: √ λ =
NCore NCore+NHalo
Agreement with prev. 1D results
]
2
[GeV/c
T
m
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 max
λ / λ
0.2 0.4 0.6 0.8 1 1.2 1.4
= 200 GeV
NN
s PHENIX 0-30% Au+Au
(syst)),
- 0.14
+0.23
0.02(stat) ± H=(0.59 /NDF=83/60, CL=2.7%
2
χ ,
2
(syst)) GeV/c
- 0.09
+0.08
0.01(stat) ± =(0.30 σ =55 MeV
- 1
' η
*=958 MeV, B
' η
m =168 MeV
- 1
' η
*=530 MeV, B
' η
m =55 MeV
- 1
' η
*=530 MeV, B
' η
m =55 MeV
- 1
' η
*=250 MeV, B
' η
m )]
2
σ )/(2
2 π
- m
2 T
1 - H exp[-(m PRL105,182301(2010), PRC83,054903(2011), resonance model: Kaneta and Xu
2
(0.55-0.9) GeV/c
〉 λ 〈 =
max
λ
- π
- π
+
π
+
π
B´ alint Kurgyis Hydrodynamics and femtoscopy 34 / 38
SLIDE 35
Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Introduction Three-dimensional L´ evy HBT Kinematic variables 3D L´ evy HBT at PHENIX
The shape of the correlation
]
2
[GeV/c
T
m
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
α
0.8 1 1.2 1.4 1.6 1.8 3D
- π
- π
3D
+
π
+
π 1D Phys. Rev. C 97, 064911
- π
- π
1D Phys. Rev. C 97, 064911
+
π
+
π PHENIX 0-30% Centrality = 200 GeV
NN
s Au+Au PH ENIX
preliminary
L´ evy exponent vs. mT Describes the shape of cor. func. Far from Gaussian (α = 2) Far from Cauchy (α = 1) Also far from 3D Ising CEP value (α ≤ 0.5) Agreement with prev. 1D results
B´ alint Kurgyis Hydrodynamics and femtoscopy 35 / 38
SLIDE 36
Summary
Hubble-flow uµ = xµ τ p = p0 τ0 τ 3+ 3
κ
n = n0 τ0 τ 3 N(S) Perturbations δuµ = δ · F(τ)g(xν)∂µS · χ(S) δp = δ · p0π(S) τ0 τ 3+ 3
κ
δn = δ · n0 τ0 τ 3 h(xν)ν(S) Accomplishments: New relativistic, accelerating, perturbative family of solutions Many possible particular solutions Measurables can be computed Outlook: Non-spherical symmetry Other S, g(xν) and h(xν) functions Another known solution
SLIDE 37
Summary
The results of this work: Acceptable fits assuming L´ evy source in 3D These results are consistent with the 1D L´ evy results
- A. Adare et al. (PHENIX), Phys. Rev. C 97, 064911 arXiv:1709.05649
α L´ evy exponent: non-Gaussian, anomalous diffusion? Scale parameter: hydro like scaling: R ∝ 1/√mT The source is not spherical
]
2
[GeV/c
T
m
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
λ
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 3D
- π
- π
3D
+π
+π 1D Phys. Rev. C 97, 064911
- π
- π
1D Phys. Rev. C 97, 064911
+π
+π PHENIX 0-30% Centrality = 200 GeV
NNs Au+Au PH ENIX preliminary ]
2[GeV/c
Tm 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R [fm] 2 4 6 8 10 12 PHENIX
- ut
) 3D
- π
- π
(
- ut
R ) 3D
+
π
+
π (
- ut
R ) 1D Phys. Rev. C 97, 064911
- π
- π
R ( ) 1D Phys. Rev. C 97, 064911
+
π
+
π R ( ]
2
[GeV/c
T
m 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 side 0-30 % Centrality ) 3D
- π
- π
(
sideR ) 3D
+π
+π (
sideR ) 1D Phys. Rev. C 97, 064911
- π
- π
R ( ) 1D Phys. Rev. C 97, 064911
+π
+π R ( ]
2
[GeV/c
T
m 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 long = 200 GeV
NN
s Au+Au ) 3D
- π
- π
(
longR ) 3D
+π
+π (
longR ) 1D Phys. Rev. C 97, 064911
- π
- π
R ( ) 1D Phys. Rev. C 97, 064911
+π
+π R (
PH ENIX preliminary ]
2
[GeV/c
T
m
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
α
0.8 1 1.2 1.4 1.6 1.8 3D
- π
- π
3D
+π
+π 1D Phys. Rev. C 97, 064911
- π
- π
1D Phys. Rev. C 97, 064911
+π
+π PHENIX 0-30% Centrality = 200 GeV
NNs Au+Au PH ENIX preliminary
Plans: Developing a 3D Coulomb correction Further info: arXiv:1810.05402, arXiv:1809.09392, arXiv:1711.05446
Thank you for your attention!
Supported by the ´ UNKP-18-1 New National Excellence Program of the Ministry of Human Capacities.
SLIDE 38
Appendix Appendix
Phase diagram of the strongly interacting matter
B´ alint Kurgyis Hydrodynamics and femtoscopy 38 / 38
SLIDE 39
Appendix Appendix
Perturbations on a standing fluid: waves
Known solution: Standing fluid uµ = (1, 0, 0, 0) p = const. n = const. Exploit these fields: ∂µuµ = 0 ∂µp = 0 uµ∂µ = ∂0 Qµν = (uµuν − gµν) Qµν∂µ = (0, ∇) Perturbed Energy equation κ∂0δp + (κ + 1)p∂µδuµ = 0 Perturbed Euler equation (κ + 1)p∂0δuν − Qµν∂µδp = 0 Wave solution for pressure ∂2
0δp = 1
κ∆δp
B´ alint Kurgyis Hydrodynamics and femtoscopy 39 / 38
SLIDE 40
Appendix Appendix
Decomposition of energy-momentum tensor
Two equations: Lorentz-orthogonal to uµ Lorentz-perpendicular to uµ Euler equation (κ + 1)puν∂νuµ = (gµν − uµuν)∂νp Energy equation κuµ∂µp + (κ + 1)p∂µuµ = 0
B´ alint Kurgyis Hydrodynamics and femtoscopy 40 / 38
SLIDE 41
Appendix Appendix
Perturbative equations
Euler equation (κ + 1)δpuµ∂µuν + (κ + 1)pδuµ∂µuν + (κ + 1)puµ∂µδuν = (gµν − uµuν)∂µδp − δuµuν∂µp − uµδuν∂µp (21) Energy equation κδuµ∂µp + κuµ∂µδp + (κ + 1)δp∂µuµ + (κ + 1)p∂µδuµ = 0 (22) Continuity equation uµ∂µδn + δn∂µuµ + δuµ∂µn + n∂µδuµ = 0 (23)
B´ alint Kurgyis Hydrodynamics and femtoscopy 41 / 38
SLIDE 42
Appendix Appendix
Perturbed equations
Euler equation: Energy equation: Continuity: ∂µδp (κ + 1)p [gµν − uµuν] = κ − 3 τκ δuν + uµ∂µδuν (24) κuµ∂µδp + 3(κ + 1) τ δp = −(κ + 1)p∂µδuµ (25) δuµnN ′(S) N(S) ∂µS + uµ∂µδn + 3δn τ + n∂µδuµ = 0 (26)
B´ alint Kurgyis Hydrodynamics and femtoscopy 42 / 38
SLIDE 43
Appendix Appendix
Solving the energy equation
Pressure perturbation δp = δ · p0 τ0 τ 3+ 3
κ π(S). (27)
Four-velocity perturbation δuµ = δ · F(τ)g(xµ)∂µS · χ(S) (28) Orthogonality satisfied (δuµuµ = 0) Energy equation χ′(S) χ(S) = − ∂µ∂µS ∂µS∂µS − ∂µS∂µ ln g(xµ) ∂µS∂µS (29) Right side is a function of S!
B´ alint Kurgyis Hydrodynamics and femtoscopy 43 / 38
SLIDE 44
Appendix Appendix
Solution of the Euler equation
Using (27) and (28) perturbations: Euler equation: π′(S) χ(S) = (κ + 1)
- F(τ)
- uµ∂µg(xµ) − 3g(xµ)
κτ
- + F ′(τ)g(xµ)
- (30)
Right side is a function of S Restriction for S, g(xµ), F(τ)
B´ alint Kurgyis Hydrodynamics and femtoscopy 44 / 38
SLIDE 45
Appendix Appendix
Solving the continuity equation
Using (28) form of perturbation The particle density perturbation δn = δ · n0 τ0 τ 3 h(xµ)ν(S) (31) Continuity equation ν(S) χ(S)N ′(S) = −F(τ)g(xµ)∂µS∂µS uµ∂µh(xµ) (32) Right side is a funciton of S Restriction for S, h(xµ), F(τ)
B´ alint Kurgyis Hydrodynamics and femtoscopy 45 / 38
SLIDE 46
Appendix Appendix
Scaling variable S = r m/τ m
Scaling variable S = rm/τ m χ(S) = S− m+1
m
- S
2 m + 1
(33) π(S) = (κ + 1)(κ − 3) κ
- π0 − m
- 1 + S− 2
m
- ,
(34) ν(S) = m2S2 S− 2
m + 1
- S− m+1
m
- S
2 m + 1
N ′(S) (35)
B´ alint Kurgyis Hydrodynamics and femtoscopy 46 / 38
SLIDE 47
Appendix Appendix
Scaling variable S = τ m/tm
Scaling variable S = τ m/tm χ(S) = S
2 m −1
- 1 − S
2 m
3
2
(36) π(S) = (κ + 1)(κ − 3) κ π0 + m
- 1 − S
2 m
(37) ν(S) = m2S2 S
2 m −1
1 − S
2 m
N ′(S) (38)
B´ alint Kurgyis Hydrodynamics and femtoscopy 47 / 38
SLIDE 48
Source function
S(x, p) = Nδ(τ − τ0)dτd3xn0 τ0 τ 3 N(S) exp −Et − xpx − ypy − zpz τT0 τ0
τ
3
κ
N(S)
- E − xpx + ypy + zpz
t
- ·
·
- 1 + δ
- − (τ + cτ
κ−3 κ
τ
3 κ )∂0Sχ(S)τ
t + + (τ + cτ
κ−3 κ
τ
3 κ )χ(S)t
Et − xpx − ypy − zpz pµ∂µS+ + (Et − xpx − ypy − zpz)(N(S)π(S) − h(x, y, z, t)ν(S)) τT0 τ0
τ
3
κ
+ + h(x, y, z, t)ν(S) N(S)
SLIDE 49
Appendix Appendix
Single-particle distribution
N(p) = Nn0E1V1(1 + P1 + P2 + P3) + Nn0E2V2(P4 + P5) (39) The newly introduced functions: E1 = exp
- −
E2 + m2 2ET0 − p2 2ETeff
- ,
V1 =
- 2πT0τ2
E
- 1 −
T0 Teff 3 E − p2 E
- 1 −
T0 Teff
- ,
(40) E2 = exp
- −
E2 + m2 2ET0 − p2 2ETeff ,δ
- ,
V2 =
- 2πT0τ2
E
- 1 −
T0 Teff ,δ 3 E − p2 E
- 1 −
T0 Teff ,δ
- .
(41) The perturbative terms are: P1 = − δ(1 + c)τ2 r1
- τ2
0 + r2 1
, P2 = δ(1 + c)τ0 E −
p2ρ2 1
- τ2
0 +r2 1
E r1 − (p2ρ2
1)
- τ2
0 + r2 1
r3
1
, (42) P3 = δ2bcκ (3 − κ)R2 r1
- τ2
0 + r2 1
3 τ0
r1 4 , P5 = − δ(τ0 + cτ0) T0 E r2 − (p2ρ2
2)
- τ2
0 + r2 2
r3
2
, (43) P4 = δ2bE
- τ2
0 + r2 2 − p2ρ2 2
˙ R0
2τ0T0
- (κ + 1)(κ − 3)
κ τ2
0 + r2 2
r2 − cκ 3 − κ τ0 r2
- τ2
0 + r2 2
3 τ0
r2 4 . (44) B´ alint Kurgyis Hydrodynamics and femtoscopy 49 / 38
SLIDE 50
Appendix Appendix
The way of finding a concrete solution
Calculate F(τ) from (9) Choose a g(xµ) Choose h(xµ) so (10) is fullfilled Choose S Calculate χ(S) Calculate π(S) Calculate ν(S) δp δuµ δn
B´ alint Kurgyis Hydrodynamics and femtoscopy 50 / 38
SLIDE 51
Appendix Appendix
Global, single particle and paircuts
Same cuts as in previous 1D Levy analysis
- A. Adare et al. (PHENIX), Phys. Rev. C 97, 064911 arXiv:1709.05649
Event selection:
Centrality: 0 − 30% z-vertex: ±30 cm
Single particle cuts:
PID: 2σ ID matching: 2σ
Paircuts:
Custom shaped cuts in ∆ϕ − ∆z distributions for EMC,DCH,TOFE,TOFW
B´ alint Kurgyis Hydrodynamics and femtoscopy 51 / 38
SLIDE 52
Appendix Appendix
Paircuts
EMCE, EMCW, DCH: ∆ϕ > ∆ϕ0 − ∆ϕ0 ∆z0 ∆z ´ es ∆ϕ > ∆ϕ1 TOFE: ∆ϕ > ∆ϕ0 − ∆ϕ0 ∆z0 ∆z TOFW: ∆ϕ > ∆ϕ0 ´ es ∆z > ∆z0 cut DCH EMC TOFE TOFW z0 ϕ0 ϕ1 z0 ϕ0 ϕ1 z0 ϕ0 z0 ϕ0 11 0.15 0.025 18 0.14 0.020 13 0.13 15 0.085
Table: Parameter values for paircuts.
B´ alint Kurgyis Hydrodynamics and femtoscopy 52 / 38
SLIDE 53
Appendix Appendix
Binning in q
Actual distribution A(q): rel. mom. dist. of pion pairs from same events
Bose-Einstein effect and others (acceptance and detector effects.)
Background distribution B(q): pion pairs from different events
No Bose-Einstein but all the others
Two particle correlation: C2(q) = A(q)/B(q) Fine resolution at low q (Bose–Einstein peak) & long enough background
Could not increase the number of bins because of memory handling
→ Two solutions:
1 Non-uniform bin width (finer at low q and wider bins as q increases) 2 mT dependent binning (BE-peak gets wider with increasing mT) The two yields the same result (included in syst. errors)
B´ alint Kurgyis Hydrodynamics and femtoscopy 53 / 38
SLIDE 54
Appendix Appendix
q-binning
Non-equal binning: ∆qi = 2 MeV, i = 1, 2, · · · 50, ∆qi−1 ·
- qi−1+∆qi−1
qi−1
1.2 , i = 51, 52, · · · 100. mT dependent binning: qmax = √mT · 325.106 MeV, hydro scaling: R ∝ 1/√mT qmax = 300 MeV in the last mT bin
B´ alint Kurgyis Hydrodynamics and femtoscopy 54 / 38
SLIDE 55
Appendix Appendix
Systematic uncertainties
B´ alint Kurgyis Hydrodynamics and femtoscopy 55 / 38
SLIDE 56
Appendix Appendix
Coulomb-correction
Charged particles → Coulomb correction K C Coulomb
2
(q, λ, α, R) = C 0
2 (q, λ, α, R) · K(q, λ, α, R)
Same Coulomb corr. used as for 1D L´ evy analysis
Very detailed numerical table calculated in advance Calculated for different q, and parameter values Spherically symmetric source
From preliminary results we see a non-spherical source ! We plan to develop a 3D Coulomb correction
B´ alint Kurgyis Hydrodynamics and femtoscopy 56 / 38