In-medium QQ potential from lattice QCD & the generalized ...

ALEXANDER ROTHKOPF - UIS

In-medium QQ potential from lattice

QCD & the generalized Gauss-law

THE 17TH INTERNATIONAL CONFERENCE ON QCD IN EXTREME CONDITIONS – JUNE 25TH 2019 – TOKYO, JAPAN

Alexander RothkopfFaculty of Science and Technology

Department of Mathematics and Physics

University of Stavanger

References:

P. Petreczky, A.R., J. Weber, NPA982 (2019) 735 & (in preparation)

D. Lafferty, A.R., arXiv:1906.00035


A challenge to theoryIN-MEDIUM QQ POTENTIAL FROM LATTICE QCD & THE GENERALIZED GAUSS-LAW

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High statistics data available for both flavors, accumulated at run2

The 17th Int. Conf. on QCD in Extreme Conditions – June 25th 2019 – Tokyo, Japan

Bottomonium: a non-equilibrium

probe of the full QGP evolution

Charmonium: a partially equilibrated

probe, sensitive to the late stages


A challenge to theoryIN-MEDIUM QQ POTENTIAL FROM LATTICE QCD & THE GENERALIZED GAUSS-LAW

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High statistics data available for both flavors, accumulated at run2


Goal: provide first principles interpretation to intricate phenomenology

Bottomonium: a non-equilibrium

probe of the full QGP evolution

Charmonium: a partially equilibrated

probe, sensitive to the late stages


Quarkonium in Heavy-Ion collisionsIN-MEDIUM QQ POTENTIAL FROM LATTICE QCD & THE GENERALIZED GAUSS-LAW

Glasma,

pre-thermalization

t～1fm

hydroexpansion

QGPt～5-10 fm

freeze-out,

hadronization

TPC~155MeV



Quarkonium in Heavy-Ion collisionsIN-MEDIUM QQ POTENTIAL FROM LATTICE QCD & THE GENERALIZED GAUSS-LAW

Glasma,

pre-thermalization

t～1fm

hydroexpansion

QGPt～5-10 fm

freeze-out,

hadronization

TPC~155MeV

production /

formationmedium interaction

partial

equilibration?freezeout

Q

Q



Selected theory goalsIN-MEDIUM QQ POTENTIAL FROM LATTICE QCD & THE GENERALIZED GAUSS-LAW

Q

Q


Properties of equilibrium QQ

Extraction of the in-medium

potential on realistic lattices

with P. Petreczky, J. Weber: NPA982 (2019) 735

with D. Lafferty arXiv:1906.00035

Novel parametrization of the

potential for use in

phenomenology



Beyond Schrödinger:

Open-quantum-systems

descr. of real-time evolution

Connecting OQS to

EFT language of potential

Real-time QQ evol. in local

thermal equilibrium

( see talk by Takahiro Miura )

Q

Q









phenomenology


( see talk by Alexander Lehmann)

QQ realtime evol.

in the initial stages

classical statistical

simulations for gluons

+

real-time NRQCD

First exploratory

steps in the glasma


Beyond Schrödinger:

Open-quantum-systems

descr. of real-time evolution

Connecting OQS to

EFT language of potential

Real-time QQ evol. in local

thermal equilibrium

( see talk by Takahiro Miura )

Q

Q









phenomenology


Quarkonium in thermal equilibrium


IN-MEDIUM QQ POTENTIAL FROM LATTICE QCD & THE GENERALIZED GAUSS-LAW

Intuition on in-medium modification from potential based studies

ρ(ω)

ω

ρ(ω)

J/ψ ψ‘

D/D thr.

ω

medium effects


Re[V](r) Im[V](r)

r r





ρ(ω)

ω

ρ(ω)

J/ψ ψ‘

D/D thr.

ω

medium effects


Re[V](r) Im[V](r)

r r





ρ(ω)

ω

ρ(ω)

J/ψ ψ‘

D/D thr.

ω

Generic prediction:

if potential picture valid,

in-medium quarkonium

becomes lighter as T

increases

medium effects


The real-time interquark potentialIN-MEDIUM QQ POTENTIAL FROM LATTICE QCD & THE GENERALIZED GAUSS-LAW

Re

lativis

tic T

>0

fie

ld th

eo

ry

Qu

an

tum

me

ch

an

ics

QCD

Dirac fields

pNRQCD

Singlet/OctetNRQCD

Pauli fields

Exploit to treat heavy quarks non-relativistically


Brambilla et.al. Rev.Mod.Phys. 77 (2005) 1423



Re

lativis

tic T

>0

fie

ld th

eo

ry

Qu

an

tum

me

ch

an

ics

QCD

Dirac fields

pNRQCD

Singlet/OctetNRQCD

Pauli fields




Matching to underlying QCD in the infinite mass limit: Wilson loop

t

x,y,zQ Q

R



Re

lativis

tic T

>0

fie

ld th

eo

ry

Qu

an

tum

me

ch

an

ics

QCD

Dirac fields

pNRQCD

Singlet/OctetNRQCD

Pauli fields


Wilson loop: potential emerges at late times




t

x,y,zQ Q

R



Re

lativis

tic T

>0

fie

ld th

eo

ry

Qu

an

tum

me

ch

an

ics

QCD

Dirac fields

pNRQCD

Singlet/OctetNRQCD

Pauli fields



∊ℂIm[V]: Laine et al. JHEP03 (2007) 054;

Beraudo et. al. NPA 806:312,2008

Brambilla et.al. PRD 78 (2008) 014017




t

x,y,zQ Q

R



Re

lativis

tic T

>0

fie

ld th

eo

ry

Qu

an

tum

me

ch

an

ics

QCD

Dirac fields

pNRQCD

Singlet/OctetNRQCD

Pauli fields



∊ℂIm[V]: Laine et al. JHEP03 (2007) 054;

Beraudo et. al. NPA 806:312,2008

Brambilla et.al. PRD 78 (2008) 014017

In this form: Minkowski time quantities and not accessible on the lattice




t

x,y,zQ Q

R


Non-perturbative evaluation of V(R)IN-MEDIUM QQ POTENTIAL FROM LATTICE QCD & THE GENERALIZED GAUSS-LAW

How to connect to the Euclidean domain: spectral functionsA.R., T.Hatsuda & S.Sasaki

PRL 108 (2012) 162001





PRL 108 (2012) 162001



Spectral Decomposition

A.R., T.Hatsuda & S.Sasaki PoS LAT2009 (2009) 162



PRL 108 (2012) 162001







PRL 108 (2012) 162001

ω

ρ☐

(R,ω

)

Γ0

ω0 well defined V(R)

if low lying Breit-

Wigner present in

Wilson loop

spectral functionFor technical details see

Y.B., A.R. PRD86 (2012) 051503



Spectral Reconstruction

In case of usual ΔW/W=10-2 statistical un-

certatinty in W□: Bayesian inference

In case of small ΔW/W<10-3 statistical un-

certatinty in W□ also Pade approximation

incorporate prior information to regularize

the inversion task (BR method)

exploit the analyticity of the

Wilson correlator to extract spectra





PRL 108 (2012) 162001

ω

ρ☐

(R,ω

)

Γ0

ω0 well defined V(R)

if low lying Breit-

Wigner present in

Wilson loop

spectral functionFor technical details see

Y.B., A.R. PRD86 (2012) 051503



Pade spectral reconstruction


In case of high statistics data: Pade approximation stable alternative to Bayes

Project imaginary frequency correlator onto rational basis functions







Generalized complex Schlessinger formula: stable P(n-1,n) approximation, decay at large ωfor purely real Schlessinger see R. Tripolt e.t al. arXiv:1801.10348







Generalized complex Schlessinger formula: stable P(n-1,n) approximation, decay at large ωfor purely real Schlessinger see R. Tripolt e.t al. arXiv:1801.10348


Appropriate continuation of the Euclidean correlator gives retarded correlator


Extracting the potential

Example: Pade based reconstructions at β=7.825 T=407MeV Nτ=12






DFT






DFT






DFT

Beware of Pade artifacts besides peak:

e.g. positivity violation, spikes

Always find well defined lowest peak:

potential picture appears viable




Latest results on the lattice potential



Lattices with dynamical u,d,s quarks

(HISQ action, HotQCD & TUMQCD)

fixed box (Ns=48 - Nτ=12, Nτ=16) &

very high statistics 4000-9000 realizations

realistic mπ~161MeV (T=151-1451MeV)

A. Bazavov et.al. PRD97 (2018) 014510, HotQCD PRD90 (2014) 094503

P. P

etr

eczky, A

.R., J

. W

eb

er,

NP

A982 (

2019)

735

Pade based extraction for Re[V] possible

Re[V] close to color singlet free energy.

Smooth transition from Cornell @ T=0

to Debye screened @ T>TC


Latest results on the lattice potential



Lattices with dynamical u,d,s quarks

(HISQ action, HotQCD & TUMQCD)

fixed box (Ns=48 - Nτ=12, Nτ=16) &

very high statistics 4000-9000 realizations

realistic mπ~161MeV (T=151-1451MeV)

A. Bazavov et.al. PRD97 (2018) 014510, HotQCD PRD90 (2014) 094503

Finite Im[V] above Tc present

P. P

etr

eczky, A

.R., J

. W

eb

er,

NP

A982 (

2019)

735

Pade based extraction for Re[V] possible

Re[V] close to color singlet free energy.

Smooth transition from Cornell @ T=0

to Debye screened @ T>TC


An improved Gauss law approach

Strategy:

αs,σ and c are vacuum prop.

and do not change with T

For use in phenomenology applications: analytic expression for Re[V] and Im[V]





Strategy:






String-like: a=+1 q=σCoulombic: a=-1 q=αs V. V. Dixit,

Mod. Phys. Lett. A 5, 227 (1990)



Strategy:







Mod. Phys. Lett. A 5, 227 (1990)

Immerse non-perturbative charge in weak coupling HTL medium: permittivity ε original idea: Y.Burnier, A.R. Phys.Lett. B753 (2016) 232 improved derivation D.Lafferty and A.R. arXiv:1906.00035



Strategy:







Mod. Phys. Lett. A 5, 227 (1990)


``



Strategy:




3 vacuum parameters and 1 temperature dependent mD fix both Re[V] and Im[V].




Mod. Phys. Lett. A 5, 227 (1990)


``


Gauss-law solution to Re[V] & Im[V]



We find an interesting connection to the classic Karsch-Mehr-Satz result:






+

Provides justification for presence of entropic term a la Satz in the string part c.f. e.g. H. Satz, EPJC75 (2015) 193 and Guo et.al. arXiv:1806.04376






+


Gauss-Law result allows to fit the lattice data even in the non-perturbative regimeD.Lafferty and A.R. arXiv:1906.00035






+


Gauss-Law result allows to fit the lattice data even in the non-perturbative regimeD.Lafferty and A.R. arXiv:1906.00035


Continuum corrected spectral functions



Need to manually correct for lattice artefacts: TC and string tension not yet physical

Pheno fit to T=0 quarkonium Continuum corrected Debye mass

Y.Burnier, O.Kaczmarek, A.R. JHEP 1512 (2015) 101







Charmonium

S-wave channel

Via Schrödinger equation for spectral functions with the Gauss-law parametrizationD.Lafferty and A.R. arXiv:1906.00035








Shifting to lower mass and broadening hierarchichally ordered with vacuum Ebind

Charmonium

S-wave channel

Via Schrödinger equation for spectral functions with the Gauss-law parametrizationD.Lafferty and A.R. arXiv:1906.00035



Towards the ψ’ to J/ψ ratio



Assume instantaneous freezeout: T>0 states

convert to real vacuum particles at around TC

Y.B

urn

ier,

O.K

aczm

are

k, A

.R.

JH

EP

1512 (

2015)

101





In-medium dilepton emission from area under

spectral resonance peaks

( to leading order )



Y.B

urn

ier,

O.K

aczm

are

k, A

.R.

JH

EP

1512 (

2015)

101










”How many vacuum states do the

in-medium peaks correspond to?”

Y.B

urn

ier,

O.K

aczm

are

k, A

.R.

JH

EP

1512 (

2015)

101








Number density: divide in-medium

by T=0 dimuon emission rate:

Y.Burnier, O. Kaczmarek, A.R.

JHEP 1512 (2015) 101





Y.B

urn

ier,

O.K

aczm

are

k, A

.R.

JH

EP

1512 (

2015)

101








Number density: divide in-medium

by T=0 dimuon emission rate:

Y.Burnier, O. Kaczmarek, A.R.

JHEP 1512 (2015) 101





ALICE Collaboration JHEP05(2016)179

T>0spectra

Y.B

urn

ier,

O.K

aczm

are

k, A

.R.

JH

EP

1512 (

2015)

101


Modeling the ratio at lower beam energy



Interpolate lattice extracted mD values with an ansatz






At small μB: perturbation theory - At large μB: dimensional arguments






HIC yields & statistical model







HIC yields & statistical model


D.L

affe

rtyan

d A

.R. a

rXiv

:1906.0

0035


Gauss-law model & bottomonium



Current state-of-the-art setup to describe bottomonium has three ingredients:

Folding together over real-time evolution gives RAA

Anisotropic

Hydro (mD(τ,x))

Glauber initial pure

state QQ prod.

VQQ in Schrödinger Eq.

Ebind and Γ(τ,x)& &

Intent: remove modelling input (up to now: color singlet free energies + pert. ImV)







Anisotropic

Hydro (mD(τ,x))


state QQ prod.




Larger Im[V] in the lattice vetted potential: stronger suppression - good at STAR







Anisotropic

Hydro (mD(τ,x))


state QQ prod.




Larger Im[V] in the lattice vetted potential: stronger suppression - good at STAR

B.

Kro

up

pa

, A

.R., M

. S

tric

kla

nd

PR

D97 (

2018)

016017

Absence of e.g. dissipative effects leads to underestimation of yields at LHC


SummaryIN-MEDIUM QQ POTENTIAL FROM LATTICE QCD & THE GENERALIZED GAUSS-LAW


Progress in the extraction of the complex in-medium potential

Quantitative insight: transition from Cornell to Debye + non-pert. Im[V]

Realistic lattices with mπ=161MeV & temperature range T=140-1450MeV

Progress in applying the lattice results to phenomenology

Improved derivation of analytic parametrization via generalized Gauss law

Lattice vetted parametrization easily included in bottomonium simulations

ψ’ / J/ψ in instantaneous freezeout scenario: good agreement with data

Pade approximation for the T>0 potential in high statistics full QCD


SummaryIN-MEDIUM QQ POTENTIAL FROM LATTICE QCD & THE GENERALIZED GAUSS-LAW

ご清聴ありがとうございましたThank you for your attention


Progress in the extraction of the complex in-medium potential

Quantitative insight: transition from Cornell to Debye + non-pert. Im[V]

Realistic lattices with mπ=161MeV & temperature range T=140-1450MeV

Progress in applying the lattice results to phenomenology

Improved derivation of analytic parametrization via generalized Gauss law

Lattice vetted parametrization easily included in bottomonium simulations

ψ’ / J/ψ in instantaneous freezeout scenario: good agreement with data

Pade approximation for the T>0 potential in high statistics full QCD


The Bayesian reconstruction



Inversion of Laplace transform required to obtain spectra from correlators






1. Nω parameters ρl >> Nτ datapoints

2. data Di has finite precision








Give meaning to problem by incorporating prior knowledge: Bayesian approachM. Jarrell, J. Gubernatis, Physics Reports 269 (3) (1996)

Bayes theorem: Regularize the naïve χ2 functional P[D|ρ] through a prior P[ρ|I]








Our prior enforces: ρ positive definite, smoothness of ρ, result independent of units

Y.Burnier, A.R.

PRL 111 (2013) 18, 182003







Have to carry out a numerical optimization problem: the most probable spectrum





Y.Burnier, A.R.

PRL 111 (2013) 18, 182003







Have to carry out a numerical optimization problem: the most probable spectrum





Convergence to unique global extremum:

partially determined by data partially by prior info

Y.Burnier, A.R.

PRL 111 (2013) 18, 182003




Quenched QCD potential



Extraction on 323xNτ anisotropic lattices with β=6.1 and ξ=4: as=0.097fm L=3.1fm

No signs for modification of Re[V] for T<TC: glueballs too heavy to interfere binding

Deconfined gluons efficiently screen Re[V], lead to a finite value of Im[V] above TC


Mock data tests with HTL & Pade (II)

Perform mock data analysis with resummed perturbative Wilson correlators (HTL)


No plateau in effective potential

Veff=-log[Wi/Wi+1]: large spectral widths

HTL correlators: Y.Burnier, A.R. PRD87 (2013) 114019













At Nτ=12, errors of dD/D=10-2 detrimental to

Im[V] but Re[V] well reconstructed






In-medium QQ potential from lattice QCD & the generalized ...

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