Charm and beauty in the deconfined plasma
from quenched lattice QCD
PRD 92 (2015) no.11, 116003, arXiv:1508.04543 [hep-lat]
JHEP 11 (2017) 206, arXiv:1709.07612 [hep-lat]arXiv:2108.13693 [hep-lat]

大野浩史
筑波大学計算科学研究センター
宇宙史研究センター2021年度
第2回構成員会議・成果報告&交流会
2021年11月26日

Outline• Introduction– Heavy flavor, spectral function• Lattice studies– Color electric correlators and spectral functions– Quarkonium correlators and spectral functions in quenched QCD(pseudo-scalar and vector channels)• Summary

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Charm and beauty in the deconfined plasma
from quenched lattice QCD

1Heavy flavor• Heavy flavor = charm & bottom
ηc, J/Ψ, χc, ηb, Υ, χb …

D, B, …

Q = c, bq = u, d, s--- qQ) are important probes to• Quarkonia (QQ) and open heavy flavors (Qq,investigate Quark-Gluon Plasma (QGP) formed in heavy ion collisions.
– Produced in the early stage of the collisions: experiencing entire evolution of QGP– A signal of QGP formation: color Debye screening in QGP→ suppression of quarkonium production T. Matsui and H. Satz, PLB 178 (1986) 416– Sequential suppression: different binding energy for different bound state→ different melting temperature

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Charm and beauty in the deconfined plasma
from quenched lattice QCD

2

What should we understand?

CMS Collaboration@QM2018

PHENIX Collaboration, PRL 98 (2007) 172301

Understanding suppression patterns
Inputs for transport models→ dissociation temperatures→ heavy quark diffusion coefficients
In-medium properties of open/hidden heavy flavors ← All encoded in spectral functions
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Charm and beauty in the deconfined plasma
from quenched lattice QCD

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Spectral function
Euclidian (imaginary time) mesonic correlation function

Spectral function
ρ(ω,p=0)/ω2

ρ(ω,p=0)/ω2

T<Tc

Ground stateρ(ω,p=0)/ω2

T>Tc

Heavy quark diffusion coefficient

T→∞(Free case)transport peak (V)transport peak(V)Exited stateω
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ωCharm and beauty in the deconfined plasma
from quenched lattice QCDωρVii : Vector spectral function
χ00 : Quark number susceptibility
4Difficulties on the lattice• Heavy quark mass mQ is too heavy!

– Fine and large lattices are needed to control lattice artifacts.

• Obtaining spectral functions: an ill-posted inverse problem!
– # of correlator data points << # of frequency bins of spectral functions– A naive χ2-fiting gives infinite number of possible spectra within statisticaluncertainties.

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Charm and beauty in the deconfined plasma
from quenched lattice QCD5Overcoming difficulties• Heavy quark mass mQ is too heavy!

– mQ >> ΛQCD → separation of scales (especially for bottom)– Effective field theories: Non-relativistic QCD (NRQCD), potential NRQCD (pNRQCD), …
Review: N. Brambilla et al., Rev. Mod. Phys. 77 (2005) 1423

• Obtaining spectral functions: an ill-posted inverse problem!
– Adding prior information– Bayesian inference: Maximum Entropy Method (MEM), Bayesian Reconstruction (BR)
Method, …
– Phenomenologically motivated (perturbative) modeling of spectral functions– Other approached without spectral functions

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Charm and beauty in the deconfined plasma
from quenched lattice QCD6

Color electric correlator• Heavy quark effective theory → color electric correlator
S. Caron-Huot, M. Laine and G.D. Moore, JHEP 04 (2009) 053• Heavy quark momentum diffusion coefficient

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Charm and beauty in the deconfined plasma
from quenched lattice QCD

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Perturbative model spectral function• Modeling the spectral function– IR partA. Francis, O. Kaczmarek, M. Laine, T. Neuhaus
and HO, PRD 92 (2015) no.11, 116003– UV part: computed from perturbation theory, two different ansatzes– Interpolations: two different interpolations, three different models

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Charm and beauty in the deconfined plasma
from quenched lattice QCD

8Heavy quark momentum diffusion coefficient
1.5Tc
Correlator in the cont. limit

Perturbatively constrainedmodel spectral functions

A. Francis, O. Kaczmarek, M. Laine, T. Neuhaus
and HO, PRD 92 (2015) no.11, 116003

Heavy quark momentum diffusion coeff.

→ 2πTD ≈ 3.7－7.0
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Charm and beauty in the deconfined plasma
from quenched lattice QCD

9Heavy quark momentum diffusion coefficient: update• Improvement with gradient flow
1.5Tc

Correlator in the cont. limit

L. Altenkort, A. M. Eller, O. Kaczmarek, L. Mazur, G. D. Moore, H.-T. Shu,
PRD 103 (2021) no.1, 014511

Perturbatively constrainedmodel spectral functions

Heavy quark momentum
diffusion coeff.

→ 2πTD ≈ 1.7－2.7
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Charm and beauty in the deconfined plasma
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Modeling quarkonium spectral functions in the pseudo-scaler channelY. Burnier, H. -T. Ding, O. Kaczmarek, A. -L. Kruse, M. Laine, HO, H. Sandmeyer, JHEP11(2017)206• High energy ρvac: Vacuum asymptotics

Burnier, Laine, Eur.Phys.J.C 72 (2012) 1902

• Threshold region ρNRQCD: pNRQCD Laine, JHEP 0705:028,2007• Suppressed at low energy

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Charm and beauty in the deconfined plasma
from quenched lattice QCD

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Fitting lattice data to the model spectral function• Quenched QCD, Clover Wilson, continuum extrapolated, pseudo-scalar
Y. Burnier, H. -T. Ding, O. Kaczmarek, A. -L. Kruse, M. Laine, HO, H. Sandmeyer, JHEP11(2017)206

Charm, Ps• The model spectral function
describes lattice data perfectly.

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Charm and beauty in the deconfined plasma
from quenched lattice QCD

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Fitting lattice data to the model spectral function• Quenched QCD, Clover Wilson, continuum extrapolated, pseudo-scalar
Y. Burnier, H. -T. Ding, O. Kaczmarek, A. -L. Kruse, M. Laine, HO, H. Sandmeyer, JHEP11(2017)206

Charm, Ps• The model spectral functiondescribes lattice data perfectly.

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Charm and beauty in the deconfined plasma
from quenched lattice QCD

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Resulting spectral functions• Quenched QCD, Clover Wilson, continuum extrapolated, pseudo-scalar
Y. Burnier, H. -T. Ding, O. Kaczmarek, A. -L. Kruse, M. Laine, HO, H. Sandmeyer, JHEP11(2017)206

Charm, Ps

Bottom, Ps• The model spectral function
describes lattice data perfectly.
• No resonance peak needed forcharm.
• A resonance peak is needed forbottom at T ≲ 1.5 Tc.
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Charm and beauty in the deconfined plasma
from quenched lattice QCD14

Extension to the vector channel• Quenched QCD, Clover Wilson, continuum extrapolated, vector
H.-T. Ding, O. Kaczmarek, A.-L. Kurse, HO, H. Sandmeyer and H.-T. Shu, arXiv:2108.13693 [hep-lat]

Transport peak is not described by ρpert
Transport peak : ω ~ 0→ τ independent contributions→ can be removed by correlator difference• The model spectral function describes
the correlator difference perfectly.
• Difference between the original latticedata and fit results→ Indication of the transport contributions
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Charm and beauty in the deconfined plasma
from quenched lattice QCD

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Resulting spectral functions• Quenched QCD, Clover Wilson, continuum extrapolated, vector
H.-T. Ding, O. Kaczmarek, A.-L. Kurse, HO, H. Sandmeyer and H.-T. Shu, arXiv:2108.13693 [hep-lat]Charm, V

Bottom, V• The model spectral function describes the correlator difference perfectly.
• No resonance peak needed for charm.
• A resonance peak is needed for bottom at T ≲ 1.5 Tc.
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Charm and beauty in the deconfined plasma
from quenched lattice QCD

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Transport contributions• Quenched QCD, Clover Wilson, continuum extrapolated, vector
H.-T. Ding, O. Kaczmarek, A.-L. Kurse, HO, H. Sandmeyer and H.-T. Shu, arXiv:2108.13693 [hep-lat]: Einstein relation• Hard to determine the transportcoefficient D

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Charm and beauty in the deconfined plasma
from quenched lattice QCD

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Transport contributions• Quenched QCD, Clover Wilson, continuum extrapolated, vector
H.-T. Ding, O. Kaczmarek, A.-L. Kurse, HO, H. Sandmeyer and H.-T. Shu, arXiv:2108.13693 [hep-lat]: Einstein relation

ωcut : frequency to separate the transport and remaining parts
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Charm and beauty in the deconfined plasma
from quenched lattice QCD

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Estimation of the transport coefficient• Quenched QCD, Clover Wilson, continuum extrapolated, vector
H.-T. Ding, O. Kaczmarek, A.-L. Kurse, HO, H. Sandmeyer and H.-T. Shu, arXiv:2108.13693 [hep-lat]

Method 1: using correlator values at the midpoint

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Method 2: using the thermal moments of the correlator…

Charm and beauty in the deconfined plasma
from quenched lattice QCD

19

Estimation of the transport coefficient• Quenched QCD, Clover Wilson, continuum extrapolated, vector
H.-T. Ding, O. Kaczmarek, A.-L. Kurse, HO, H. Sandmeyer and H.-T. Shu, arXiv:2108.13693 [hep-lat]

Method 1: using correlator values at the midpoint

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Method 2: using the thermal moments of the correlator…

Charm and beauty in the deconfined plasma
from quenched lattice QCD

20

Summary and outlook• Heavy quarks are important probes to investigate quark-gluon-plasma.
• Spectral functions contain all information about in-medium properties ofheavy quarks.
• There are some difficulties to study the spectral functions on the lattice.
• Heavy quark momentum diffusion coefficient was estimated by fitting thecolor electric correlator to a perturbative model spectral function.
• Charmonium and bottomonium spectral functions were determined fromcontinuum extrapolated quenched lattice correlators in pseudo-scalar andvector channels• Heavy quark diffusion coefficient was also estimated.
• Extension to full QCD is a future work.
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Charm and beauty in the deconfined plasma
from quenched lattice QCD21