NEW EXACT AND PERTURBTIVE SOLUTIONS OF RELATIVISTIC HYDROpyweb.swan.ac.uk/~aarts/talks-thor-lisboa/Csanad_THOR... · 2018-06-14 · new exact and perturbtive solutions of relativistic

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


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]


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


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


7/27

TIME EVOLUTION OF PERTURBATIONS

• An example perturbation from the class of solutions:


June 13, 2018

density n(x) pressure p(x) flow u(x)


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



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


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


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


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?


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


14/27

PSEUDORAPITY DENSITIES IN A+A

• Described well from RHIC to LHC

Jiang, Yang, MCs, Csörgő, Phys. Rev. C 97, 064906, 2018


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?


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


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


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


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



20/27

TEMPERATURE EVOLUTION

• Recall: limited domain in 𝜂, where (𝜂 𝐻 , Ω H ) relation functional

• Strong dependence on EoS, analytic understanding



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



22/27

COMPARISON TO DATA

• Description valid in limited 𝜂 interval only

• Result very close to CNC solution



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.


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?


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


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


http://zimanyischool.kfki.hu/18

Ceterum censeo

Carthaginem esse delendam

… if you are interested in these subjects, come to:

THANK YOU FOR

YOUR ATTENTION!

27/27

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NEW EXACT AND PERTURBTIVE SOLUTIONS OF RELATIVISTIC HYDROpyweb.swan.ac.uk/~aarts/talks-thor-lisboa/Csanad_THOR... · 2018-06-14 · new exact and perturbtive solutions of relativistic