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Quarkonium dissociation and regeneration

Juhee HongYonsei University

Phys. Rev. C 99, 034905 (2019) JH, Su Houng LeePhys. Lett. B 801, 135147 (2020) JH, Su Houng Lee

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Simulation for Heavy IoN Collision with Heavy-quark and ONia(SHINCHON)

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Quark-gluon plasma (QGP)• Early universe before 10-6 s after the Big Bang• Relativistic heavy ion collisions

• Heavy quarks created in hard processes at the beginning, remain frozen

dissociation, regeneration hadronic effects

T0 ⁓ 550 MeV TC ⁓ 192 MeV TF ⁓ 115 MeV

quark-gluon

plasma

hadron gas

pre-equilibrium hadronization freeze-out

t0 ⁓ 0.3fm/c tC⁓ 7fm/c tF⁓ 17fm/c

hadronsg, q, ϒ

the expansion rate > the scattering rate

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Quarkonium

• Bound states of a heavy quark (b, c) and its antiquark

• Quark-gluon plasma formation, thermal properties

• Color screening

• Interaction length ↔ bound state radius

• Melting at Tdiss

H. Satz (2006)

T. Matsui, H. Satz (1986)

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Quarkonium suppression

• Nuclear modification factor (RAA)

RAA ⁓ 𝑁𝐴𝐴

ϒ

𝑁𝑐𝑜𝑙𝑙

𝑁𝑝𝑝

ϒ

• Quarkonium dissociation by in-medium interaction

• Quarkonium regeneration

• Feed-down

• Cold nuclear matter effects

CMS Collaboration (2019)

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Finite temperature QCD

• Heavy quark-antiquark static potential: screened Coulomb

• The imaginary part of the singlet potential: thermal width

• Quarkonium dissociation by scattering processes

- Gluo-dissociation (LO) - Inelastic parton scattering (NLO)

M. Laine, O. Philipsen, M. Tassler, P. Romatschke (2007)N. Brambilla, J. Ghiglieri, A. Vairo, P. Petreczky (2008)

hard thermal loop (HTL) resummation

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g≪ 1 for high Thard ⁓ Tsoft ⁓ gT

perturbative

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Outline

• Quarkonium dissociation in a partonic picture:

Gluo-dissociation, inelastic parton scattering

- Numerical analysis for ϒ(1S)

• Quarkonium regeneration

Inverse gluo-dissociation, inelastic parton scattering

• Boltzmann equation for quarkonium transverse momentum spectra:

Nuclear modification factor RAA, elliptic flow v2

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Quarkonium dissociation

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Gluo-dissociation

• g+ϒ →

• Interactions between partons and HQ bound state:dipole interaction of color charge with gluon

• Bethe-Salpeter amplitude in ϒ rest frame:

q, p1,p2»k

G. Bhanot, M. E. Peskin (1979)

Y. Oh, S. Kim, S. H. Lee (2002)

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• Coulombic bound state:

the relative momentum p=(p1-p2)/2

• Bohr radius:

• Dissociation cross section:

• Thermal width:

k≥E

n(k)=1

𝑒𝑘/𝑇−1

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• LO: k,k0 »mD

• NLO transverse dispersion for k«T: k02 - k2 - ReПT =0

HTL self-energy in Coulomb gauge:

• NLO cross section:

agrees with potential nonrelativistic QCD (pNRQCD) for mv » T » E » mDpNRQCD: m » mv » E (nonrelativistic)

T » mD (thermal)

N. Brambilla, M. A. Escobedo, J. Ghiglieri, A. Vairo (2013)

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the imaginary part using the octet propagator

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Inelastic parton scattering• p + ϒ → p + (p=g, )

• Effective vertex:

• ϒ dissociation at NLO:

T. Song, S. H. Lee (2005)

→+

gluo-dissociation at LO

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k0≈0 for small energy transfer ⁓ the singlet-octet chromoelectric

dipole vertex in pNRQCD

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• Small energy transfer for weakly coupled quarkonia: k10 ≈ k20

• Hard thermal loop (HTL) propagator:

• Dissociation cross section:

• Collinear breakup:

• Thermal width:

N. Brambilla, M. A. Escobedo, J. Ghiglieri, A. Vairo (2013)

agrees with pNRQCD for mv » T » mD » E

T ⁓

⁓mD

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nF(k1)=1

𝑒𝑘1/𝑇+1, nB(k1) =

1

𝑒𝑘1/𝑇−1

small anglescattering

Landau damping

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Higher order corrections

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O(g) corrections

• Higher order expansion with the effective vertex

• Same order for soft(⁓gT) gluons:

• Bose enhancement in the presence of soft gluons: αsn(k) ⁓ g2 T/k ⁓ g

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gT ⁓ gT ⁓

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⁓ g2

⁓ 1

𝑔2

4π

1

𝑒𝑘/𝑇 − 1

T ⁓hard

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Heavy quark diffusion

• Momentum diffusion coefficient of a HQ:the mean-squared momentum transfer per unit time

• Thermal width:

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bath particles

G. D. Moore, D. Teaney (2005)

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• κNLO with soft gluons

• κNLO is useful to obtain O(g) corrections to ΓNLO: ΓO(g) ⁓ κNLO

•Other O(g) corrections induced by heavy quark-antiquark interactions

S. Caron-Huot, G. D. Moore (2008)

ΓNLO with hard partons

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Numerical analysis for ϒ(1S)

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• ϒ(1S) survives up to Tdiss⁓ 600MeV

• Transitional behavior in QGP at Tc ≤ T ≤ Tdiss

• Binding energy from lattice QCD

• Debye screening mass: 𝑚𝐷2 =

𝑔2𝑇2

3𝑁𝑐 +

𝑁𝑓

2

A. Mocsy, P. Petreczky (2007)

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ϒ(1S)

Tc

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Numerical calculations

• Gluo-dissociation:

- NLO slightly smaller

• Inelastic parton scattering by numerical integral over the phase space:

- depends on binding energy

- approach the asymptotic formula

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E

log

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Dissociation cross section

• Numerical integration over the phase space:

• Gluo-dissociation peak depending on EInelastic parton scattering decreases with large mD at high T

• Low k1: gluo-dissociation is efficientHigh k1: inelastic parton scattering is dominant

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Thermal width

• Phase space: larger at high T, peak at higher k

• Gluo-dissociation/inelastic parton scattering is dominant at low/high T• Total thermal width increases with T: dissociation when Γtot ≥ E• Larger width at weaker αs (smaller mD, larger a0)

E»T tunnellingD. Khareev et al. (1995)

21⁓ 1/g2⁓ gT

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Quarkonium moving through QGP

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Kinetic approach

• Rate equation with absorption and production:

𝑑𝑁

𝑑𝑡= −Γ𝑁 + (production)

• Boltzmann equation:

𝜕

𝜕𝑡+ 𝑣 ∙

𝜕

𝜕𝑥𝑓 𝑝

= −∫𝑝2,𝑝3,𝑝4𝑀 2 𝑓𝑓2 1 ± 𝑓3 1 ± 𝑓4 − 𝑓3𝑓4 1 ± 𝑓 1 ± 𝑓2

× (2π)4δ4(𝑝 +𝑝2 −𝑝3−𝑝4)

p

p2

p3

p4

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Hadronic models

P. Braun-Munzinger, J. Stachel (2000)V. Greco, C. M. Ko, R. Rapp (2004)

• Recombination by hadronic processes near the phase transition

• Statistical hadronization: 𝑁𝑒𝑞 = 𝑔γ𝑉 ∫𝑑3𝑝

2𝜋 3𝑓(𝑝)

• Coalescence: 𝑑2𝑁

𝑑2𝑝𝑇⁓ ∫𝑑2𝑝1𝑇𝑑

2𝑝2𝑇𝑑2𝑁𝑏

𝑑2𝑝1𝑇

𝑑2𝑁ഥ𝑏𝑑2𝑝2𝑇

δ2( Ԧ𝑝𝑇 − Ԧ𝑝1𝑇 − Ԧ𝑝2𝑇)

• Continuous regeneration by the Boltzmann equation

(dof)(fugacity)(volume)

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thermal widththermal widththermal width regenerationregeneration

Quarkonium transport

b

b−ϒ b

b−ϒ

• Boltzmann equation with dissociation and regeneration

• Dissociation:

• Regeneration by b+b → ϒ+g , b+b+p → ϒ+p

• Quarkonium regeneration depends on HQ density, Nϒ/NQ

Nϒ/Nb « NJ/ψ/Nc

− −

⁓ 10-2σpp→ϒ/σpp→b ⁓ 10-3 ⁓ 10-2

ϒ velocity = q/q0

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Thermal width for ϒ(1S) moving in QGP

• Calculate the scattering amplitudes in the ϒ rest frame

• Partons move w.r.t. ϒ(1S): f k =1

𝑒𝑝·𝑢/𝑇±1, 𝑢μ = γ 1, 𝑣 , γ =

1

𝑣2−1

• Thermal width by convoluting the parton distributions with the cross section

• Thermal width at the plasma rest frame, dividing by the Lorentz factor

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thermal bath velocity

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• Higher q: smaller thermal width by gluo-dissociationlarger width by inelastic parton scattering

• The total thermal width increases with ϒ(1S) momentum

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Quarkonium regeneration

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Quarkonium regeneration

• Inverse gluo-dissociation, inelastic parton scattering

• Regeneration depends on HQ distribution function

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b

b− b

b−

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Heavy quark distribution

• Boltzmann equation with elastic scattering (p+b → p+b)

• Small energy transfer: Fokker-Planck equation

• Langevin equation: 𝑑𝑥

𝑑𝑡=

𝑝

𝑚

• Random walk suffering many collisions

• HQ approach to the equilibrium slowly by ⁓ m/T

⁓mD

drag momentum diffusion

B. Svetitsky (1988)H. van Hees, R. Rapp (2005)

G. D. Moore, D. Teaney (2005)

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⁓ 𝑇

𝑚suppressed

random momentum kicks

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• Heavy quark distribution

• At leading-log order (vHQ=0):

• Einstein relation: spatial diffusion

• Green function:

• Solution:

initial: differential cross section in pp collisions

H. van Hees, R. Rapp (2005) G. D. Moore, D. Teaney (2005)

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Thermal medium evolution

• Viscous hydrodynamics:

• A longitudinal motion by a Bjorken expansion

- ideal gas equation of state: e=3P ⁓ T4

- local equilibrium at t0=0.3 fm/c with T0=550 MeV

T ≤ 7fm for QGP

R ⁓ 7fm

ideal

shear viscosityenergy density, pressureequation of state

J. D. Bjorken (1983)

⁓ T2

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• Conservation of bottom: fugacity by a statistical model (Nϒ«Nb)

• Evolution of b quarks:

⁓ 1

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Heavy quark distribution

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Regeneration term

• Inverse gluo-dissociation is efficient at low TInelastic parton scattering is dominant at high T

• Higher q: smaller regeneration by inverse gluo-dissociationlarger regeneration by inelastic parton scattering

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• Regeneration is stronger for smaller diffusion

• Inverse gluo-dissociation is sensitive to D at low T

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ϒ(1S) transverse momentum spectra

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Dynamic evolution of ϒ(1S)• In the central rapidity region with the Lorentz boost invariance

• Initial distribution: differential cross section in pp collisions

• Dominant dissociation: Nϒ reduces to 40%• Small fϒ at high qT: regeneration effects

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Nuclear modification factor

• Tf ≈ Tc at tf ≈ 7fm/c:

• Low qT: suppression by dissociationHigh qT: suppression, enhancement by regeneration

• Less suppression for smaller D

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• Significant uncertainties in the initial stage of heavy ion collisions

• RAA curves depend on b,ϒ initial distributions:

• Softer spectrum with larger α, smaller Λ: stronger regeneration at high qT

• Higher T0: longer QGP lifetime, smaller RAA

Dependence on initial conditions

, Λb=6.07GeV

, Λϒ=6.05GeV, α=2.44

softer bsofter ϒ

harder bharder ϒ

⁓ smaller diffusion

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•Higher energy collisions with larger T0: smaller RAA

• Significant regeneration effects on RAA at high qT

• Sum of gluo-dissociation and inelastic parton scattering: RAA⁓ 0.4

• RAA data appear to be independent of qT: suppression by dissociation and enhancement by regeneration

CMS Collaboration (2017, 2019)

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Comparison to experiment

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Feed-down

State Tdiss (MeV)

ϒ(1S) 593

ϒ(2S) 228

ϒ(3S) < 192

Χb1 265

Χb2 < 192

A. Mocsy, P. Petreczky, M. Strickland (2013)

feed-down fractions

ϒ(1S) → ϒ(1S) 0.668

ϒ(2S) → ϒ(1S) 0.086

ϒ(3S) → ϒ(1S) 0.010

χb(1P) → ϒ(1S) 0.170

χb(2P) → ϒ(1S) 0.051

χb(3P) → ϒ(1S) 0.015

B. Krouppa, A. Rothkopf, M. Strickland (2018)

• Inclusive RAA = σ𝑠 𝑓𝑠𝑅𝐴𝐴

𝑠 Dissociation temperatures

• Gluo-dissociation is important for excited states

• If 𝑅𝐴𝐴2𝑆 ⁓ 𝑅𝐴𝐴

1𝑃, 𝑅𝐴𝐴3𝑆 , 𝑅𝐴𝐴

2𝑃, 𝑅𝐴𝐴3𝑃 ∶

feed-down effects ⁓ 10% of 𝑅𝐴𝐴1𝑆

CMS Collaboration (2019)

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Elliptic flow

• 𝑣2 =𝑝𝑥2−𝑝𝑦

2

𝑝𝑥2+𝑝𝑦

2

• Azimuthal angular anisotropy: b quark v2

• Elliptic flow induced by regeneration:

•With the same initial conditions, RAA and v2 are comparable to experiment

M. He, R. J. Fries, R. Rapp (2017)

ALICE Collaboration (2019)PNU

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y

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Other approaches

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Langevin simulationJ. P. Blaizot, D. De Boni, F. Faccioli, G. Garberoglio (2016)

• Dynamics of heavy quarks including bound state formation and dissociation

• Generalized Langevin equation with the force and noise determined from correlation functions of the equilibrium plasma

• Dissociation by a screened potential and collisions with partons, regeneration when enough heavy quarks are present

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Open quantum system

S. Kajimoto, Y. Akamatsu, M. Asakawa, A. Rothkopf (2018)

• Quarkonium dynamics by a Schrodinger equation with an in-medium potential and noise terms describing HQ-medium interactions

• Noise correlation: wave decoherence

• For short lcorr , the initial ground state is easily excited and mixed with excited states

noise

inelastic parton scattering

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Summary

• Quarkonium dissociation and regeneration by scattering processes:

For collinear breakup with small energy transfer, NLO dissociation cross sections agree with pNRQCD

• Thermal width increases with T, qRegeneration depends on HQ diffusion: stronger regeneration for smaller D

• ϒ(1S) transverse momentum spectra by the Boltzmann equation

• Nuclear modification factor: suppression by dissociation at low qTsuppression and enhancement by regeneration at high qT

• Elliptic flow induced by regeneration

• With the same initial conditions, both RAA and v2 are comparable with the CMS, ALICE data

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Thank you for your attention!

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