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NEW EXACT AND PERTURBTIVESOLUTIONS OF RELATIVISTIC HYDROA COLLECTION OF RECENT RESULTS
MÁTÉ CSANÁD (EÖTVÖS U) @ THOR LISBON MEETING, JUNE 13, 2018
+T. CSÖRGŐ, G. KASZA, Z. JIANG, C. YANG, B. KURGYIS, M. NAGY, …
PHASES OF QUARK MATTER
• An evolution throughout many phases
• Modeling possible with hydrodynamics (?)
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
2/27
EXACT HYDRO HISTORY & BASICS
• Relativistic hydrodynamics: established by Landau (for p+p!)
• Exact, analytic solutions important: connect initial and final state
• Famous solutions by Landau&Khalatnikov and 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
See e.g. this review: de Souza, Koide, Kodama, Prog. Part. Nucl. Phys. 86, 35 (2016)
• Analytic solutions capture many features of data
MCs, Vargyas, Eur. Phys. J. A 44, 473 (2010)
MCs, Szabo, Phys. Rev. C 90, 054911 (2014)
• Still lacking: non-spherical 3D, accelerating, realistic solutions
• Linearized hydro: perturbations
Kurgyis, MCs, Universe 3 (2017) no.4, 84
Shi, Liao and Zhuang, Phys.Rev. C90 (2014) no.6, 064912
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
3/27
A NEW CHALLENGE
• PHENIX observing Lévy sources:
ℒ 𝛼, 𝑅; 𝑟 =1
2𝜋 3 𝑑3𝑞𝑒𝑖𝑞𝑟𝑒−
1
2𝑞𝑅 𝛼
• Shape parameter: Gauss if 𝛼 = 2,
power-law tail if 𝛼 < 2
• How to reconcile with hydro?
• Exponential cutoff exp −𝑝𝜇𝑢𝜇/𝑇
in Boltzmann-Jüttner?
• Rescattering?
June 13, 2018
Gauss (α=2.0) Lévy (α=1.2)
M. Csanád (Eötvös U) @ THOR Lisbon Meeting
Cauchy
Lévy (a=1.2)
Gauss
Log
sourc
e d
ensi
ty
Distance
4/27
THE PERTURBATIVE METHOD
• Method: perturbed equations for a known solution
• Linearized hydro equations:
• Need a specific „base”
solution
• Example: standing fluid
𝜅𝜕0𝛿𝑝 + 𝜅 + 1 𝑝𝜕𝜇𝛿𝑢𝜇 = 0
𝜅 + 1 𝑝𝜕0𝛿𝑢𝜇 − 𝑄𝜇𝜈𝜕𝜇𝛿𝑝 = 0
with 𝑄𝜇𝜈 = 𝛿𝜇1𝛿𝜈1 − 𝑔𝜇𝜈
• Result: waves
𝜕02𝛿𝑝 = 𝑐𝑠
2Δ𝑝
Method similar to Shi, Liao and Zhuang
Phys.Rev. C90 (2014) no.6, 064912 [arXiv:1405.4546]
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
5/27
A NEW CLASS OF PERTURBATIVE SOLUTIONS
• Hubble-flow: 𝑢𝜇 =𝑥𝜇
𝜏, 𝑝 = 𝑝0
𝜏0
𝜏
3+3
𝜅, 𝑛 = 𝑛0
𝜏0
𝜏
3𝒩(𝑠), 𝑢𝜇𝜕𝜇𝑠 = 0
• Describes observables, including HBT and higher order flow
MCs, Szabo, Phys. Rev. C 90, 054911 (2014),
MCs, Vargyas, Eur. Phys. J. A 44, 473 (2010)
• Perturbative solution on top of Hubble-flow possible:
Kurgyis, MCs, Universe 3 (2017) no.4, 84, arXiv:1711.05446
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
6/27
RESCRITIONS FOR PERTURBATIVE SOLUTIONS
• Flow profile 𝜒(𝑆), pressure profile 𝜋(𝑆), density profile 𝜈(𝑆)
• Auxiliary functions 𝐹 𝜏 , 𝑔 𝑥𝜈 , ℎ 𝑥𝜈
• These are related to each other as:
• Left side: only depends on scale variable 𝑆!
• Many solutions possible, various scaling variables and profiles
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
7/27
TIME EVOLUTION OF PERTURBATIONS
• An example perturbation from the class of solutions:
Kurgyis, MCs, Universe 3 (2017) no.4, 84, arXiv:1711.05446
June 13, 2018
density n(x) pressure p(x) flow u(x)
M. Csanád (Eötvös U) @ THOR Lisbon Meeting
8/27
PERTURBATIONS OF THE OBSERVABLES
• Observables calculable via usual Jüttner-Boltzmann source w/ Cooper-Fry
• Spectra and correlations obtain a perturbative component
• Hubble-flow observables stable against small perturbations
Kurgyis, MCs, Universe 3 (2017) no.4, 84, arXiv:1711.05446
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
9/27
THE BJORKEN-ESTIMATE
• The original idea: energy density based on dE/dy
• QGP critical 𝜖𝑐 ~ 1 GeV/fm3 (from 𝜖𝑐 = (6 − 8)𝑇𝑐4)
• Result (~2000x cited)
• Boost invariant flow
Phys.Rev. D27 (1983)
• Needs correction!
𝐸 = 𝑁𝑑𝐸
𝑑𝑦Δ𝑦 = 𝑁
𝑑𝐸
𝑑𝑦
1
2
2𝑑
𝑡= 𝜖𝐴𝑑
𝜖Bj =1
𝑅2𝜋𝜏0
𝑑𝐸
𝑑𝜂=
𝐸
𝑅2𝜋𝜏0
𝑑𝑁
𝑑𝜂
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
10/27
AN ANALYTIC SOLUTION WITH ACCELERATION
• The CNC solution in 1 + 𝑑 dimensionsCsörgő, Nagy, Csanád, Phys.Lett. B663 (2008) 306-311
Nagy, Csörgő, Csanád, Phys.Rev. C77 (2008) 024908
𝑣 = tanh 𝜆𝜂, 𝑝 = 𝑝0𝜏0
𝜏
𝜆𝑑𝜅+1
𝜅cosh
𝜂
2
− 𝑑−1 𝜙𝜆
𝜎 = 𝜎0𝜈 𝑠𝑝
𝑝0
𝜅
𝜅+1, 𝑇 =
𝑇0
𝜈 𝑠
𝑝
𝑝0
𝜅
𝜅+1
• Classes of solutions:
June 13, 2018
𝝀 𝒅 𝜿 𝝓𝝀
Hwa-Bjorken 1 ∈ ℝ ∈ ℝ 0
Fixed acceleration, any dim. 2 ∈ ℝ 𝑑 0
𝒅 = 𝟏, 𝜿 = 𝟏, any acceleration ∈ ℝ 𝟏 𝟏 𝟎
Fixed deceleration 1/2 ∈ ℝ 1 (𝜅 + 1)/𝜅
Fixed acceleration 3/2 ∈ ℝ (4𝑑 − 1)/3 (𝜅 + 1)/𝜅
M. Csanád (Eötvös U) @ THOR Lisbon Meeting
11/27
AN ADVANCED ENERGY DENSITY ESTIMATE
• Fact: 𝑑𝑁/𝑑𝑦 not flat
• Finiteness & acceleration
• Acceleration parameter l
• Corrections needed:
• 𝑦𝜂 & 𝜂final 𝜂initial
• Work done by pressure
• Corrected estimate for 𝜅 = 1
𝜖 = 𝜖Bj 2𝜆 − 1𝜏𝑓
𝜏𝑖
𝜆−1
, 𝜏 = 𝜆𝜏Bj = 𝜆𝑚𝑇
𝑇𝑓𝑅long
• Björken estimate: only for 𝜅 = ∞ (dust EoS)
• Will come back to this soon
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
12/27
THE PSEUDORAPIDITY DENSITY FROM CNC
•𝑑𝑁
𝑑𝑦≅ 𝑁0 cosh
−𝛼
2−1 𝑦
𝛼exp −
𝑚
𝑇𝑓cosh𝛼
𝑦
𝛼
• Main parameter: 𝛼 =2𝜆−1
𝜆−1
• Particle mass 𝑚,
• Freeze-out temp. 𝑇𝑓
• Measure acceleration
from rapidity distributions
• Extension to more
complex flows?
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
13/27
INITIAL ENERGY DENSITY AT RHIC
• Bjorken estimate from BRAHMS: 𝜖Bj =𝐸
𝑅2𝜋𝜏0
𝑑𝑁
𝑑𝜂≅ 5 GeV/fm3
• Advanced estimate: 𝜖 = 𝜖Bj 2𝜆 − 1 𝜏𝑓/𝜏𝑖𝜆−1
• Correction: 2-3x, result ~15 GeV/fm3, QCD agreement!
• Corresponds to Tini 2Tc 340 MeV, confirmed by g spectra
June 13, 2018
BRAHMS
dN/dη
M. Csanád (Eötvös U) @ THOR Lisbon Meeting
14/27
PSEUDORAPITY DENSITIES IN A+A
• Described well from RHIC to LHC
Jiang, Yang, MCs, Csörgő, Phys. Rev. C 97, 064906, 2018
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
15/27
ENERGY DENSITIES IN AA, RHIC TO LHC
• Effect of acceleration and conjectured effect of Equation of State
Jiang, Yang, MCs, Csörgő, Phys. Rev. C 97, 064906, 2018
• Effect of EoS: important to understand analytically!
• What about p+p?
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
16/27
BJORKEN ENERGY DENSITY ESTIMATE IN PP
• Rough estimate via the Bjorken formula:𝜖Bj =𝐸
𝑅2𝜋𝜏0
𝑑𝑁
𝑑𝜂• Number of particles at midrapidity: 1.5 × 5.89
• Average energy: 𝑚𝑡 = 𝐸 = 0.562 GeV
• Transverse size of the system R2𝜋 = 𝜎tot2 /4𝜎el = 9.8 fm2
• Formation time 𝜏0 = 1 fm/𝑐 (conservative estimate)
• Energy density from this:
𝜖Bj 7 TeV =1
𝑅2𝜋𝜏0
𝑑𝐸
𝑑𝜂=
𝐸
𝑅2𝜋𝜏0
𝑑𝑛
𝑑𝜂=0.562 × 1.5 × 5.89
1.762𝜋
GeV
fm3 = 0.507GeV
fm3
𝜖Bj 8 TeV =1
𝑅2𝜋𝜏0
𝑑𝐸
𝑑𝜂=
𝐸
𝑅2𝜋𝜏0
𝑑𝑛
𝑑𝜂=0.571 × 1.5 × 6.17
1.802𝜋
GeV
fm3 = 0.519GeV
fm3
MCs, Csörgő, Jiang, Yang, Universe 3 (2017) no.1, 9, arXiv:1609.07176
• This is at average multiplicity; compare to 𝜖crit ≈ 1GeV
fm3
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
17/27
ENERGY DENSITY IN P+P
• Data from p+p well described
• 7 TeV: 𝜆 = 1.073, 𝜖corr = 0.645GeV
fm3
• 8 TeV: 𝜆 = 1.067, 𝜖corr = 0.641GeV
fm3
• 13 TeV: 𝜆 = 1.065, 𝜖corr = 0.692GeV
fm3
• Multiplicity dependence
• Energy density above 1 GeV/fm3 for multiplicites above ~10! EoS dependence?
MCs, Csörgő, Jiang, Yang, Universe 3 (2017) no.1, 9, arXiv:1609.07176 + manuscript in preparation
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
18/27
A NEW CLASS OF EXACT SOLUTIONS
• How to reconcile the Bjorken estimate (𝜅 = ∞) with hydro?
• Search for new solutions with flow 𝑢𝜇 = coshΩ 𝜂 , sinhΩ 𝜂
• Energy and Euler equations become:
𝜕𝜂Ω + 𝜅 𝜏𝜕𝜏 + tanh Ω − 𝜂 𝜕𝜂 ln 𝑇 = 0
𝜕𝜂 ln 𝑇 + tanh Ω − 𝜂 𝜏𝜕𝜏 ln 𝑇 + 𝜕𝜂Ω = 0
• A new class of solutions emerges, if one relaxes self-similarity
• These will be implicit, introducing
𝜂 𝐻 = Ω 𝐻 − 𝐻
Ω 𝐻 =𝜆
𝜆 − 1 𝜅 − 𝜆atan
𝜅 − 𝜆
𝜆 − 1tanh𝐻
Csörgő, Kasza, MCs, Jiang, Universe 2018, 4(6), 69 arXiv:1805.01427
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
19/27
THE NEW CLASS OF SOLUTIONS
𝑢𝜇 = coshΩ 𝐻 , sinhΩ 𝐻
𝜎 𝜏,𝐻 = 𝜎0𝜏0𝜏
𝜆
𝜈 𝑠 1 +𝜅 − 1
𝜆 − 1sinh2𝐻
−𝜆/2
𝑇 𝜏,𝐻 = 𝑇0𝜏0𝜏
𝜆𝜅 1
𝜈 𝑠1 +
𝜅 − 1
𝜆 − 1sinh2𝐻
−𝜆/2𝜅
𝑠 𝜏, 𝐻 =𝜏0𝜏
𝜆−1
sinh𝐻 1 +𝜅 − 1
𝜆 − 1sinh2𝐻
−𝜆/2
• Quantities given parametrically as (𝜂 𝐻 , Ω H )
• Simplification: limit the solution in 𝜂 where 𝜂 → Ω univalent (functional)
• Not self-similar: Coordinate dependence not only via scaling variable 𝑠
• Explicit and exact solution
Csörgő, Kasza, MCs, Jiang, Universe 2018, 4(6), 69 arXiv:1805.01427
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
20/27
TEMPERATURE EVOLUTION
• Recall: limited domain in 𝜂, where (𝜂 𝐻 , Ω H ) relation functional
• Strong dependence on EoS, analytic understanding
Csörgő, Kasza, MCs, Jiang, Universe 2018, 4(6), 69 arXiv:1805.01427
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
21/27
OBSERVABLES
• Rapidity density calculable in saddle-point approximation
𝑑𝑁
𝑑𝑦≅ 𝑁0 cosh
−𝛼(𝜅)2
−1 𝑦
𝛼(1)exp −
𝑚
𝑇𝑓cosh𝛼 𝜅
𝑦
𝛼 1− 1
where 𝛼 𝜅 =2𝜆−𝜅
𝜆−𝜅was introduced
• Normalization:
𝑁0 =𝑅2𝜋𝜏𝑓
2𝜋ℏ 3
2𝜋T𝑓𝑚3
𝜆(2𝜆 − 1)exp −
𝑚
𝑇𝑓
• Pseudorapidity density as
parametric 𝜂 𝑦 →𝑑𝑁
𝑑𝜂𝑦 curve
• Using Jacobian: 𝑑𝑦
𝑑𝜂=
𝑝𝑡 𝑦 cosh 𝜂 𝑦
𝑚2+ 𝑝𝑡 𝑦2 cosh2 𝜂 𝑦
and 𝑝𝑡 𝑦 =𝑇𝑓2+2𝑚𝑇𝑓
1+𝛼 𝜅
2𝛼 1
𝑇𝑓+𝑚
𝑇𝑓+2𝑚𝑦2
June 13, 2018
𝑑𝑁
𝑑𝑦
𝑑𝑁
𝑑𝜂
M. Csanád (Eötvös U) @ THOR Lisbon Meeting
Csörgő, Kasza, MCs, Jiang, Universe 2018, 4(6), 69 arXiv:1805.01427
22/27
COMPARISON TO DATA
• Description valid in limited 𝜂 interval only
• Result very close to CNC solution
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
Csörgő, Kasza, MCs, Jiang, Universe 2018, 4(6), 69 arXiv:1805.01427
23/27
WHAT ABOUT THE ENERGY DENSITY?
• Bjorken estimate: 𝜖Bj =𝐸
𝑅2𝜋𝜏0
𝑑𝑁
𝑑𝜂
• Valid only for dust EoS, 𝜅 = ∞
• CNC solution, finiteness and acceleration (only these effects), valid for 𝜅 = 1
• Correction factor: 2𝜆 − 1𝜏𝑓
𝜏0
𝜆−1
• Work done by pressure (without acceleration, just the expansion)
• Correction factor: 𝜏𝑓
𝜏0
𝜆
• CKCJ solution, exact EoS dependent result
• Correction factor: 2𝜆 − 1𝜏𝑓
𝜏0
1+1
𝜅𝜆−1
• Energy density: 𝜖 =𝑑𝑁
𝑑𝜂
𝐸
𝑅2𝜋𝜏02𝜆 − 1
𝜏𝑓
𝜏0
1+1
𝜅𝜆−1
Csörgő, Kasza, WPCF2018&private comm.
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
24/27
DETERMINING THE INITIAL STATE
• Dependence on EoS: from direct photons and/or lattice QCD
• Dependence on multiplicity: plug in measured value
• Dependence on final/initial time: largest source of uncertainty
• What about the effect of viscosity?
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
Kasza, WPCF2018 & private comm.
25/27
A NEW VISCOUS SOLUTION
• A new analytic solution with bulk viscosity
• 𝑢𝜇 =𝑥𝜇
𝜏
• 𝑛 = 𝑛0𝜏0
𝜏
3𝜈 𝑠
• 𝑝 = 𝑝0𝜏0
𝜏
3𝜅+1
𝜅+
𝜁
𝜅
3
𝜏
3
3𝜅+1
𝜅−1
• 𝑇 = 𝑇0𝜏0
𝜏
3
𝜅+
𝜁
𝜅𝑛0
3
𝜏
𝜏
𝜏
3𝒯(𝑠)
3
3𝜅+1
𝜅−1
• Viscous heating at late stages
• Note: shear viscosity cancels for Hubble-flow!
• New shear viscous analytic solutions in preparation
Jiang, Yang, Csörgő, Kasza, Nagy, MCs, in preparation
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
26/27
SUMMARY
• Many new results in exact/analytic hydro
• Perturbations on top of Hubble-flow
• Allows to introduce complicated anisotropies
• To be expanded to other solutions
• Advanced energy density estimates
• Björken estimate: no acceleration, no pressure
• Advanced estimates based on exact solutions
• High energy densities reached in LHC p+p
• New accelerating families of solutions
• Arbitrary acceleration, arbitrary EoS
• EoS dependent energy density estimate
June 13, 2018M. Csanád (Eötvös U) @ THOR Lisbon Meeting
http://zimanyischool.kfki.hu/18
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YOUR ATTENTION!
27/27
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