[PPT] - Charmonium spectral functions from 2+1 flavour lattice QCD asztor 12 PowerPoint Presentation

SLIDE 1

Charmonium spectral functions from 2+1 flavour lattice QCD

Attila P´ asztor 12 apasztor@bodri.elte.hu

In collaboriation with

S. Bors´

anyi3

S. D¨

urr34

Z. Fodor134
C. Hoelbling3
S. D. Katz12
S. Krieg34
S. Mages5
D. N´
gr´

adi1

A. Sch¨

afer5

B. C. T´
th3
N. Trombit´

as1

K. K. Szab´
34

1E¨

tv¨
s University

2MTA-ELTE Lattice Gauge Theory Group 3University of Wuppertal 4J¨

ulich Supercomputing Center

5University of Regensburg

BGZ Triangle Meeting, 2014, Balatonf¨ ured

SLIDE 2

Motivation

We will consider correlators of charmonium currents in the

pseudoscalar(PS) and vector(V) channels, i.e. ¯ cγ5c and 3

i=1 ¯

cγic

These correspond to the ηc and J/Ψ mesons
J/Ψ suppression is regarded as an important signal of QGP

formation

Charmonium states are regarded as a possible thermometer

for the QGP

This study is with 2+1 dynamical quarks, previous studies:

mostly quenched, one in 2 flavour The talk is based on: JHEP 1404 (2014) 132

SLIDE 3

Preliminaries

Spectral function ∼ im part of real-time retarded correlator

A(ω) = (2π)2 Z

m,n
e−En/T − e−Em/T

|n|JH(0)|m|2 δ(p − kn + km) Quasipaticle ∼ peaks in the SF, melting/dissociation ∼ peaks not well separated anymore, they become part of the continuum

Relation to the Euclidean time correlator

G(τ, p) = ∞ dωA(ω, p)K(ω, τ) where K(ω, τ) = cosh(ω(τ − 1/2T)) sinh(ω/2T) Left hand side is calculable with lattice techniques. This is very hard to invert.

SLIDE 4

The Maximum Entropy Method

The method in a nutshell

Q = αS − 1 2χ2 S =

dω
A(ω) − m(ω) − A(ω) log

A(ω) m(ω)

χ2 =
i,j

(G fit

i

− G data

i

)C −1

ij (G fit j

− G data

j

) Gi =

A(ω)K(ω, τi)dω

m(ω) is a function, summarizing our prior knowledge of the

solution. Then we average over α. The conditional probability

P[α|data, m] is given by Bayes’ theorem.

SLIDE 5

The Maximum Entropy Method

Conclusions from mock data analysis

The peak positions OK, the shapes not so much
With not too noise data points, O(10) points are OK
Features that remain unchanged by varying the prior are

restricted by the data

Resultion becomes worse at bigger ω
Peaks close in position can be merged into one broader peak.

Take home message: the stable thing is just the position of the first (ground state)

SLIDE 6

Simulation details

Lattice details

Gauge action = Symanzik tree-level improved gauge action Fermion action = 2+1 dynamical Wilson fermions with 6 step stout smearing (ρ = 0.11) and tree-level clover improvement Same configurations as in JHEP 1208 (2012) 126 a = 0.057(1)fm mπ = 545MeV Ns = 64 Nt = 28...12 T = 123...288MeV We measured the charm meson correlators on these lattices.

SLIDE 7

Outline of MEM procedure

Stability test at the lowest temp

Drop data points, emulating the number of data points

available at the lowest temperature (Nt = 28)

Do the same analysis as with the higher temperature
correlators. If the ground state peak cannot be reconstructed,

the given number of data points is not reliable

RESULT: Nt=12 NOT OK, Nt=14,16,18,20 OK

Error analysis

Systematic error analysis: vary ∆ω, Nω, the shape of the prior

function: m0, m0ω2, 1/(m0 + ω), m0ω and m0=0.01, 0.1, 1.0, 10.0.

Statistical error analysis: given set of parameters, 20 jackknife

samples

SLIDE 8

Sensitivity on prior function

This is the PS channel, but V looks similar

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.5 1 1.5 2 2.5 3 3.5 A(ω)/ω2 ωa PSEUDOSCALAR CHANNEL, Nt=20, T/Tc≈0.9 m(ω)=0.01/a2 m(ω)=0.1ω2 m(ω)=1.0/a2/(10.0+ωa)

SLIDE 9

Temperature dependence

0.05 0.1 0.15 0.2 0.25 0.5 1 1.5 2 2.5 3 3.5 A(ω)/ω2 aω PSEUDOSCALAR CHANNEL, m(ω) = 0.1 ω2 Nt=14, T/Tc ≈ 1.30 Nt=16, T/Tc ≈ 1.14 Nt=18, T/Tc ≈ 1.01 Nt=20, T/Tc ≈ 0.91

SLIDE 10

Temperature dependence

Vector channel

0.05 0.1 0.15 0.2 0.25 0.5 1 1.5 2 2.5 3 3.5 A(ω)/ω2 aω VECTOR CHANNEL, m(ω) = 0.1 ω2 Nt=14, T/Tc ≈ 1.30 Nt=16, T/Tc ≈ 1.14 Nt=18, T/Tc ≈ 1.01 Nt=20, T/Tc ≈ 0.91

SLIDE 11

Results - MEM

Pseudoscalar channel - position of 1st peak

0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 Position of first peak in lattice units aT=1/Nt

SLIDE 12

Results - MEM

Vector channel - position of 1st peak

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 Position of first peak in lattice units aT=1/Nt

SLIDE 13

Charm diffusion coefficient

Kubo-formula

D = 1 6χ lim

ω→0 3

i=1

Aii(ω, T) ω , If D > 0 we have ρ/ω > 0 for small ω implying a transport peak

Narrow transport peak

In the case of a narrow transport peak, we can use the ansatz: Atransport(ω, T) = f (T)ωδ(ω − 0+) This does not mean, that the diffusion coefficient is infinite. But in case of a narrow transport peak, the Euclidean correlator G(τ, T) =

K(ω, τ)A(ω, T) is not sensitive to the full shape of

the peak, only the area. The contrubtion of the transport peak will be a temperature dependent constant (zero mode).

SLIDE 14

Some indication of a transport peak

Nt = 16 not conclusive

0.003 0.006 0.009 0.025 0.05 A(ω)/(ωT) ωa m(ω)=0.01 a2 m(ω)=a-2/(10+ω a) m(ω)=0.1ω2 m(ω)=1.0ω/a m(ω)=0.01ω/a

SLIDE 15

A different method

Definition of G/Grec

Datta, Karsch, Petreczky, Wetzorke 2004

G (t, T) Grec (t, T) = G(t, T)

A(ω, Tref)K(ω, t, T)dω

Midpoint subtracted version

G − G −

rec

= G (t, T) − G (Nt/2, T) Grec (t, T) − Grec (Nt/2, T) = G (t, T) − G (Nt/2, T)

A(ω, Tref) [K(ω, t, T) − K(ω, Nt/2, T)] dω

This removes the zero mode.

SLIDE 16

Results: G/Grec

Pseudoscalar channel

0.94 0.96 0.98 1 1.02 1.04 1.06 2 4 6 8 10 G/Grec t/a Nt=12 Nt=14 Nt=16 Nt=18 Nt=20

SLIDE 17

Results: G/Grec

Vector channel

0.9 0.95 1 1.05 1.1 1.15 2 4 6 8 10 G/Grec t/a Nt=12 Nt=14 Nt=16 Nt=18 Nt=20

SLIDE 18

Results: G −/G −

rec

Pseudoscalar channel, midpoint subtracted version

0.9 0.95 1 1.05 1.1 2 4 6 8 10 G-/G-

rec

t/a Nt=12 Nt=14 Nt=16 Nt=18 Nt=20

SLIDE 19

Results: G −/G −

rec

Vector channel, midpoint subtracted version

0.9 0.95 1 1.05 1.1 2 4 6 8 10 G-/G-

rec

t/a Nt=12 Nt=14 Nt=16 Nt=18 Nt=20

SLIDE 20

Results: G − Grec

Vector channel

1e-05
5e-06

5e-06 1e-05 2 4 6 8 10 G-Grec t/a Nt=16 Nt=14 Nt=12

SLIDE 21

Conclusions

MEM analysis

There seems to be no change in the SF in the PS channel up

to 1.4Tc

There seems to be some change in SF in the V channel
Indications of a transport peak in the V channel

G/Grec analyis

No change in the PS channel
In the V channel, results are consistent with a temperature

independent ω > 0 part and a temperature dependent zero mode (narrow transport peak), described by the ansatz A(ω) = f (T)ωδ(ω − 0+) + A(ω, T = 0).

SLIDE 22

Backup - implementation details 1

MEM continued...

It can be shown, that the maximum of Q is in an Ndata dimensional subspace: A(ω) = m(ω) exp Ndata

i=1

sifi(ω)

Two choice for basis functions: Bryan (Eur. Biophys J. 18, 165

(1990)) or Jakov´ ac et al (Phys.Rev. D75 014506 (2007). We use the latter. In this case the maximization of Q is equivalent to the minimization of U = α 2

Ndata

i,j=1

siCijsj + ωmax dωA(ω) −

Ndata

i=1

Gdata

i

si. Comment: Have to use arbitrary precision arithmetics with both methods.

SLIDE 23

Backup - implementation details 2

Problem: stopping criterion

0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25

2000 4000 6000 8000 10000 12000 14000 16000 U # of iteration steps

SLIDE 24

Backup - implementation details 3

Solution: going back to the Nω dimensions

5.95 6 6.05 6.1 6.15 6.2 6.25 6.3 5000 10000 15000 20000 25000 30000 35000 40000

Q

# of iteration steps

SLIDE 25

Backup - charm mass tuning

From Davies et al PRL 104, 132003 (2010) mc/ms = 11.85. Because of additive renormalization, it is impossible to use this

directly. We use (mc − ms)/(ms − mud) where the additive

renormalization constant cancels. We know that for ud and s the masses used in the simulation correspond to a mass ratio of 1.5 (Durr et al. Phys. Lett. B701 (2011) 265), from this we get (mc − ms)/(ms − mud) = 32.55 We check if the meson masses are indeed in the right ballpark: JP mi ma ma/mD∗

s a

mexp/mD∗

s