Color screening in 2+1 flavor QCD, testing EQCD and spectral functions

Color screening in 2+1 flavor QCD, testing EQCD and spectral functions

Free energy of static quark anti-quark at T>0 and deconfinement transition in 2+1 flavor QCD

Onset of color screening and EQCD

Spatial string tension and further tests of EQCD

Spectral functions and heavy quark diffusion constant

Deconfinement and color screening in QCD

The free energy of static quark anti-quark pair agrees with the T=0 potential for r << 1/T The free energy of static quark anti-quark pair is screened for r>r

_scr

at any temperature r

_sc

r ~ 1/T => Debye screening

2+1 flavor QCD, continuum extrapolated, TUMQCD, PRD 98 (2018) 054511

-4 -3 -2 -1 0

F

_QQ^_

(r,T)-T ln 9 [GeV]

r [fm]

T [MeV]

0

1600 1200800 500 320 260 220 180 160 140

0.01 0.02 0.03 0.06 0.1 0.2 0.3 0.6

0.01 0.02 0.03 0.06 0.1 0.2 0.3 0.6

Deconfinement and color screening in QCD (cont’d)

Pure glue ≠ QCD !

Deconfinement transition happens at lower temperature but the Polyakov loop behaves smoothlyaround T

_c

, Z(3) symmetry plays no apparent role

How to define deconfinement transition in QCD ?

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

150 200 250 300 350 400 450 500 T [MeV]

L

_ren

(T)

stout, cont.

HISQ, cont.

SU(3)

SU(2)

Polyakov and gas of static-light hadrons

Energies of static-light mesons:

Megias, Arriola, Salcedo, PRL 109 (12) 151601

Bazavov, PP, PRD 87 (2013) 094505

Ground state and first excited states are from lattice QCD

Michael, Shindler, Wagner, arXiv1004.4235

Wagner, Wiese, JHEP 1107 016,2011

Higher excited state energies

are estimated from potential model

Gas of static-light mesons only works for T < 145 MeV

100 200 300 400 500 600

120 140 160 180 200

T [MeV]

F

_Q

(T) [MeV] HISQ

stout

Free energy of an isolated static quark:

1

2

The entropy of static quark

S

_Q

= @F

_Q

@ T

The onset of screening corresponds to peak is S

_Q

and its position coincides with T

_c

1 1.5 2 2.5 3 3.5 4 4.5 5

140 160 180 200 220 240 260 280 300 T [MeV]

S_Q

T_χ(N_τ)

N_τ=6 local fit global 1/N_τ⁴ fit HRG

1 1.5 2 2.5 3 3.5 4 4.5 5

120 140 160 180 200 220 240 260 280 300 T [MeV]

S_Q

T_χ(N_τ)

N_τ=8 local fit global 1/N_τ⁴ fit global 1/N_τ² fit HRG

1 1.5 2 2.5 3 3.5 4 4.5 5

120 140 160 180 200 220 240 260 280 300 T [MeV]

S_Q

T_χ(N_τ)

N_τ=10 local fit global 1/N_τ⁴ fit global 1/N_τ² fit HRG

1 1.5 2 2.5 3 3.5 4 4.5 5

120 140 160 180 200 220 240 260 280 300 T [MeV]

S_Q

T_χ(N_τ)

N_τ=12 local fit global 1/N_τ⁴ fit global 1/N_τ² fit HRG

TUMQCD, PRDD9 3 (2 01 6) 1 14 50 2

The entropy of static quark

S

_Q

= @F

_Q

@ T

At low T the entropy S

_Q

increases reflecting the increase of states the heavy quark can be coupled to; at high temperature the static quark only “sees” the medium within a Debye radius, as T increases the Debye radius decreases and S

_Q

also decreases

The peak in the entropy is broader and smaller for smaller quark mass Weak coupling (EQCD) calculations work for T> 1500 MeV

Berwein et al, PRD 93 (2016) 034010

0 1 2 3 4 5 6 7 8 9 10

0.8 1 1.2 1.4 1.6 1.8 2 S_Q

T/T_c

N_f=2+1, m_π=161 MeV N_f=3, m_π=440 MeV N_f=2, m_π=800 MeV N_f=0

T [MeV]

S_Q(T) LO

NLO, µ=(1-4)π T NNLO lattice

0 0.1 0.2 0.3 0.4 0.5

1000 1500 2000 2500 3000 3500 4000 4500 5000

TUMQCD, PRDD93 (2016) 114502

Free energy of a static quark anti-quark pair at high T

Conjecture: in QCD the work is reduced due to cancelation of color singlet and octet contribution

e

^FQQ¯(r,T)/T

= 1

9 e

^FS(r,T)/T

+ 8

9 e

^FO(r,T)/T

Fierz idenity:

_{ij lk}

= 1

N

_c ^{ik lj}

+ 2T

_ik^a

T

_lj^a

e

^FS(r,T)/T

= 1

N

_c

h trW (r)W

^†

(0) i , W (r ) =

N

Y

⌧ 1

⌧=0

U

₀

(r, ⌧ )

The work to separate the Q Q ¯ pair from distance r

₁

to r

₂

: F

_QQ¯

(r

₂

) F

_QQ¯

(r

₁

)

LO:

The LO result for F

_QQ¯

is recovered F

_S

= 4

3

↵

_s

r e

^mDr

+ F

₁

, F

_O

= + 1 6

↵

_s

r e

^mDr

+ F

₁

Leading order in perturbation theory:

F

_QQ¯

(r, T ) = 1 9

↵

²_s

r

²

T exp( 2m

_D

r) + F

₁

In QED at leading order:

F

_QQ¯

(r, T ) = ↵

r exp( m

_D

r) + F

₁

Free energy of a static quark anti-quark pair at high T (cont’d)

The definition of F

_S

requires gauge fixing Is it possible to have a gauge invariant decomposition of F

_QQ¯

into singlet and octet contributions ?

Yes, for r ⌧ 1/T using pNRQCD

L

_A

is Polyakov loop in the adjoint representation e

^FQQ¯(r,T)

= h S (r, 1/T )S (r, 0) i + L

_A

h O

^a

(r, 1/T )O

^a

(r, 0) i

e

^FQQ¯(r,T)

= 1

9 e

^fs(r,T)/T

+ 8

9 e

^fo(r,T)/T

, f

_s

= V

_s

(r) + O (↵

²_s

rT

²

), f

_o

= V

_o

(r) N

_c

↵

_s

m

_D

2 + O (↵

²_s

rT

²

)

Brambilla et al, PRD 82 (2010) 074019

Berwein et al, PRD 96 (2017) 014025 The naïve and the pNRQCD decomposition into singlet and octet agree

To calculate F

_QQ¯

(r, T ) and F

_S

(r, T ) for r ⇠ 1/m

_D

another EFT, namely EQCD, should be used

.

F

_s

= f

_s

+ O (↵

_s³

T ) + O (↵

²_s

rT

²

), F

_O

= f

_o

+ O (↵

³_s

T ) + O (↵

²_s

rT

²

)

<latexit sha1_base64="5eAUhcr55jqMutvPFEV+BGK13Vc=">AAACW3iclVFLSwMxGMyuj9b1VRVPXoJFaFHK7iroRRCE4q0KrS102+XbNNuGZh8kWaEs9Ud60oN/RUxrDz56cSAwzHzDl0yClDOpbPvNMFdW19YLxQ1rc2t7Z7e0t/8ok0wQ2iIJT0QnAEk5i2lLMcVpJxUUooDTdjC+nfntJyokS+KmmqS0F8EwZiEjoLTkl0Tdl9ehL09zjwDHjWnFA56OwJf9c9ysLpFdLJp9t3r2XPcbOpn8O+mXynbNngP/Jc6ClNEC937pxRskJItorAgHKbuOnapeDkIxwunU8jJJUyBjGNKupjFEVPbyeTdTfKKVAQ4ToU+s8Fz9nsghknISBXoyAjWSv72ZuMzrZiq86uUsTjNFY/K1KMw4VgmeFY0HTFCi+EQTIILpu2IyAgFE6e+wdAnO7yf/JY9uzdH84aJ84y7qKKIjdIwqyEGX6AbdoXvUQgS9og+jYBSNd3PFtMytr1HTWGQO0A+Yh5/wTbFT</latexit><latexit sha1_base64="5eAUhcr55jqMutvPFEV+BGK13Vc=">AAACW3iclVFLSwMxGMyuj9b1VRVPXoJFaFHK7iroRRCE4q0KrS102+XbNNuGZh8kWaEs9Ud60oN/RUxrDz56cSAwzHzDl0yClDOpbPvNMFdW19YLxQ1rc2t7Z7e0t/8ok0wQ2iIJT0QnAEk5i2lLMcVpJxUUooDTdjC+nfntJyokS+KmmqS0F8EwZiEjoLTkl0Tdl9ehL09zjwDHjWnFA56OwJf9c9ysLpFdLJp9t3r2XPcbOpn8O+mXynbNngP/Jc6ClNEC937pxRskJItorAgHKbuOnapeDkIxwunU8jJJUyBjGNKupjFEVPbyeTdTfKKVAQ4ToU+s8Fz9nsghknISBXoyAjWSv72ZuMzrZiq86uUsTjNFY/K1KMw4VgmeFY0HTFCi+EQTIILpu2IyAgFE6e+wdAnO7yf/JY9uzdH84aJ84y7qKKIjdIwqyEGX6AbdoXvUQgS9og+jYBSNd3PFtMytr1HTWGQO0A+Yh5/wTbFT</latexit><latexit sha1_base64="5eAUhcr55jqMutvPFEV+BGK13Vc=">AAACW3iclVFLSwMxGMyuj9b1VRVPXoJFaFHK7iroRRCE4q0KrS102+XbNNuGZh8kWaEs9Ud60oN/RUxrDz56cSAwzHzDl0yClDOpbPvNMFdW19YLxQ1rc2t7Z7e0t/8ok0wQ2iIJT0QnAEk5i2lLMcVpJxUUooDTdjC+nfntJyokS+KmmqS0F8EwZiEjoLTkl0Tdl9ehL09zjwDHjWnFA56OwJf9c9ysLpFdLJp9t3r2XPcbOpn8O+mXynbNngP/Jc6ClNEC937pxRskJItorAgHKbuOnapeDkIxwunU8jJJUyBjGNKupjFEVPbyeTdTfKKVAQ4ToU+s8Fz9nsghknISBXoyAjWSv72ZuMzrZiq86uUsTjNFY/K1KMw4VgmeFY0HTFCi+EQTIILpu2IyAgFE6e+wdAnO7yf/JY9uzdH84aJ84y7qKKIjdIwqyEGX6AbdoXvUQgS9og+jYBSNd3PFtMytr1HTWGQO0A+Yh5/wTbFT</latexit><latexit sha1_base64="5eAUhcr55jqMutvPFEV+BGK13Vc=">AAACW3iclVFLSwMxGMyuj9b1VRVPXoJFaFHK7iroRRCE4q0KrS102+XbNNuGZh8kWaEs9Ud60oN/RUxrDz56cSAwzHzDl0yClDOpbPvNMFdW19YLxQ1rc2t7Z7e0t/8ok0wQ2iIJT0QnAEk5i2lLMcVpJxUUooDTdjC+nfntJyokS+KmmqS0F8EwZiEjoLTkl0Tdl9ehL09zjwDHjWnFA56OwJf9c9ysLpFdLJp9t3r2XPcbOpn8O+mXynbNngP/Jc6ClNEC937pxRskJItorAgHKbuOnapeDkIxwunU8jJJUyBjGNKupjFEVPbyeTdTfKKVAQ4ToU+s8Fz9nsghknISBXoyAjWSv72ZuMzrZiq86uUsTjNFY/K1KMw4VgmeFY0HTFCi+EQTIILpu2IyAgFE6e+wdAnO7yf/JY9uzdH84aJ84y7qKKIjdIwqyEGX6AbdoXvUQgS9og+jYBSNd3PFtMytr1HTWGQO0A+Yh5/wTbFT</latexit>

/T

/T

Free energy at short distances: lattice results vs. pNRQCD

-20 -10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 V_S(r)-F_S(r,T) [MeV]

rT

Continuum limit

-20 -10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Final value

-20 -10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -20

-10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -20

-10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

N_τ≥6, O₄ /= 0

-20 -10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

N_τ≥8, O₄= 0

-20 -10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

N_τ

-20 -10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

12

-20 -10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

10

-20 -10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

8

-20 -10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

6

-20 -10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

4

-20 -10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -20

-10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -20

-10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -20

-10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -20

-10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -20

-10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -20

-10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -20

-10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -20

-10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -20

-10 0 10 20 30 40 50 60

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.006 0.008 0.01 0.012 0.014 0.016

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 r² TF_QQ^_(r,T)

rT 0.006

0.008 0.01 0.012 0.014 0.016

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

fit data uncor.

α_s³ cor.

The di↵erence between V

_s

and F

_S

is small as expected for rT < 0.3

works ! Construct pNRQCD prediction for F

_QQ¯

using the lattice data for F

_S

and V

_s

as proxy for f

_s

together with the relation:

f

_o

= 1

8 f

_s

+ 3↵

³_s

8r

✓ ⇡

²

4 3

◆

The interaction of static Q and ¯ Q is vacuum like for rT < 0.3

T=407 MeV

TUMQCD, PRD 98 (2018) 054511

Free energy in the screening regime: lattice vs. weak coupling

1e-3 1e-2 1e-1

0.2 0.4 0.6 0.8 1 1.2

rT

T [MeV]

1e-3 1e-2 1e-1

0.2 0.4 0.6 0.8 1 1.2

-9r² TF_QQ^_(r,T)

421 5814

1e-4 1e-3 1e-2 1e-1 1e0

0.5 1 1.5 2

rT

T [MeV]

1e-4 1e-3 1e-2 1e-1 1e0

0.5 1 1.5 2

T [MeV]

-rF_S(r,T)/C_F

220 333 421 1222 2923 5814

TUMQCD, PRD 98 (2018) 054511

_ _

Coulomb gauge

Lattice results are in reasonable agreement with NLO weak coupling result for rT<0.6,

at larger distances, non-pertubtative effects (due to chromo-magnetic sector ) become important F

_S

(r, T ) =

4↵

_s

e

^mDr

3r (1 + ↵

_s

( Z

₁

(µ) + rT f

₁

(rm

_D

))) F

_QQ¯

(r, T ) =

↵

²_s

e

^2mDr

9r

²

T (1 + ↵

_s

( Z

₁

(µ) + rT f (rm

_D

)))

NLO in EQCD results are available for ¯ F

_i

(r, T ) = F

_i

(r, T ) F

₁

(T )

Nadkarni, PRD 33 (1986) 3738 Burnier et al, JHEP 01 (2010) 054

Spatial string tension at T>0 and dimensional reduction

non-perturbative

Calculated perturbatively

Laine, Schröder, JHEP0503(05) 067 Cheng et al., PRD 78 (2008) 034506

EQCD :

magnetic screening

r₀ =0.469fm

Pressure and screening mass in the 3d effective theory

10⁰ 10¹ 10² 10³

T / T_c 0.0

0.5 1.0 1.5 2.0

p / T4

Stefan-Boltzmann law O(g⁶ln

-

^{1g )} + fitted O(g⁶) full EQCD + fitted O(g⁶) 4d lattice data

10⁰ 10¹ 10² 10³

T / T_c 0.0

0.5 1.0 1.5 2.0 2.5 3.0

(e - 3p) / T4

Stefan-Boltzmann law O(g⁶ln

-

^{1g )} + fitted O(g⁶) full EQCD + fitted O(g⁶) 4d lattice data

Figure 6: Left: the best fit of our result to 4d lattice data for

p/T⁴

at

N^f

= 0 [28]. Our result comes out as a function of

T /Λ_MS

; for converting to

T /T^c

we scanned the interval

T^c/Λ_MS

= 1.10

. . .

1.35 (dark band). For comparison, with the light band we show the outcome for

F_MS^R ≡

0 [29]. Right: the corresponding trace anomaly, (e

−

3p)/T

⁴

=

T

d(p/T

⁴

)/dT .

physical stripe of Fig. 1. Comparing with the lower order terms in Eq. (A.1), Eq. (4.2) also has a reasonable magnitude, and indicates that perturbation theory within EQCD is in fact a useful tool at all parameter values corresponding to the 4d theory. In principle, the numerical result could also be compared with improved resummation methods defined within EQCD (see, e.g., refs. [31]–[33]).

At the same time, from the phenomenological point of view, our result is somewhat of a disappointment: as shown in Fig. 6, the match to 4d lattice data in the range T /T

c

≥ 3.0 is not particularly smooth. In fact, as shown in the figure, including the newly determined ≥ 5- loop remainder decreases the quality of the fit significantly. This implies that, unfortunately, we are not in a position to realize our original goal of offering consolidated crosschecks for the N

f

̸ = 0 QCD pressure in the interesting temperature range T /T

c

∼ 1.5 . . . 3.0.

On the other hand, our study raises the theoretical question of what kind of effects could be responsible for the mismatch between our results and that of 4d lattice simulations. On the mundane side, one possibility would be a substantial contribution from the condensate denoted by ∂

x

F

MS

= ⟨(Tr [ ˆ A

²₀

])

²

⟩

a

− ... . Unfortunately its systematic inclusion would require a significant amount of new analytic and numerical work; moreover, as order-of- magnitude estimates and previous preliminary simulations suggest, it appears unlikely that this condensate could significantly change the qualitative behaviour that we have observed.

On a more adventurous note, let us point out that, qualitatively, the reason for the mis- match is that the condensate in Eq. (4.1) is too large, and grows rapidly as y (or T ) de- creases (note that for N

f

= 0, the range T /T

c

= 10

⁰

...10

³

corresponds to y ≃ 0.3...1.2, or

10 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

1 10 100 1000 10000

m

E

/T

T/T

_c

FIG. 16. The electric screening masses in units of the temperature. Shown are the electric masses m

^E

for the first (filled circles) and the second (open circles) set of h. The line represent the fit for the temperature dependence of the electric mass from 4d simulations. The open triangles are the values of the electric mass for h values obtained from perturbative reduction in the metastable region (the last column in Table III). Some data points at the temperature T ∼ 70T

^c

and T ∼ 9000T

^c

have been shifted in the temperature scale for better visibility.

As one can see from the figure the agreement between the masses obtained from 4d and 3d simulation is rather good. The electric mass shows some dependence on h. For relatively low temperatures (T < 50T

_c

) the best agreement with the 4d data for the electric mass is obtained for values of h corresponding to 2-loop dimensional reduction and lying in the metastable region. This fact motivated our choice of the parameters of the effective theory at T = 2T

_c

in section III. For higher temperatures, however, practically no distinction can be made between the three choices of h.

Before closing the discussion on the choice of the parameters of the effective 3d theory let us compare our procedure of fixing the parameters of the effective theory with that proposed in [9]. In Ref. [9] the gauge coupling was fixed by matching the data on Polyakov loop correlators determined in lattice simulation to the corresponding value calculated in lattice perturabtion theory. The resulting gauge couplings turned out to be considerably smaller than the corresponding ones used in our analysis, e.g. for T = 2T

_c

it gave g

²

= 1.43 while our procedure gives g

²

(2T

_c

) = 2.89.

The scalar couplings were fixed according to 1-loop perturbation theory [9]. Using this procedure the authors of Ref.

[9] obtained a good description of the Polyakov loop correlator in terms of the 3d effective theory, however, the spatial Wilson loop whose large distance behavior determines the spatial string tension was not well described in the reduced 3d theory. The reason for this is the fact that the value of the Polyakov loop correlator is cut-off dependent and therefore it is not very useful for extracting the renormalized coupling.

[1] D.J. Gross, R.D. Pisarski and L.G. Yaffe, Rev. Mod. Phys. 53 (1981) 43 [2] F. Karsch, A. Patk´ os and P. Petreczky, Phys. Lett B401 (1997) 69

[3] J.O. Andersen, E. Braaten and M. Strickland, Phys. Rev. Lett. 83 (1999) 2139; Phys.Rev. D 61 (2000) 014017; Phys.Rev.

D 61 (2000) 074016

[4] J.-P. Blaizot, E. Iancu and A. Rebhan, Phys. Rev. Lett. 83 (1999) 2906; Phys.Lett. B470 (1999) 181 [5] R. Kobes, G. Kunstatter and A. Rebhan, Phys. Rev. Lett. 64 (1990) 2992; Nucl. Phys. B355 (1991) 1 [6] A. Rebhan, Nucl. Phys. B430 (1994) 319

[7] P. Arnold and L.G. Yaffe, Phys. Rev. D52 (1995) 7208

[8] F. Karsch, M. Oevers and P. Petreczky, Phys. Lett. B442 (1998) 291 [9] P. Lacock, D.E. Miller and T. Reisz , Nucl. Phys. B369 (1992) 501

[10] L. K¨ arkk¨ ainen, P. Lacock, B. Petersson and T.Reisz Nucl.Phys. B395 (1993) 733

[11] L. K¨ arkk¨ ainen P. Lacock, D.E. Miller, B. Petersson and T.Reisz, Nucl. Phys. B418 (1994) 3 [12] K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B503 (1997) 357 [13] M. Laine and O. Philipsen, Phys. Lett. B459 (1999) 259

[14] A. Hart and O. Philipsen, Nucl.Phys. B572 (2000) 243

17 Pressure in EQCD

Kajantie et al, PRD79 (2009) 045018

Electric screening mass from gluon propagator

Cucchieri et al, PRD64 (2001) 036001

Line is the fit to the 4d lattice results.

Different symbols correspond to

different choices of the 3d mass

parameter

) , ( )

, ( )

, (

, ) 0 , 0 ( )

, ( )

, ,

(

³

x x

x J

J x J

e x d T

p G

H H

H H

x p i

t y t

y t

t t

G

=

= ò

^{× +}

Euclidean correlators and spectral functions

µ

µ

g g

g

g ×

=

G

_H

1 ,

_{5, , 5}

) (

) ,

( t T D i t

G =

^>

-

ò

^¥

-

=

0

sinh( /( 2 ))

)) 2 /(

1 (

cosh(

) , ( )

,

( T

T T d

T

G w

t w w

ws t

) ( ) ( 1 Im

2

) ( )

( w s w

p p

w

w -

^<

= =

>

D

R

D D

Lattice QCD is formulated in imaginary time

Physical processes take place in real time

) , ( )

0 , 0 ( )

, , (

, ) 0 , 0 ( )

, ( )

, , (

3 3

x t J J

e x d T

p t D

J x t J e

x d T

p t D

H H

x p i

H H

x p i

ò ò

×

<

×

>

=

=

if

fit the large distance behavior of the lattice correlation functions

This is not possible for Maximum Entropy Method (MEM)

Current-current correlators and heavy quark diffusion

⇢

^µ⌫_V

(!) ⌘

Z

₁

1

dte

^i!t

Z

d

³

x D

[ ˆ J

^µ

(t, ~ x), J ˆ

^⌫

(0, ~ 0)] E

drag constant

X

i

⇢

ⁱⁱ_V

(!)

! ' 3

^q₂

D ⌘

²

⌘

²

+ !

²

, ! < !

_{U V}

, ⌘ = T M

1 D

Spatial diffusion constant ~ mean free path (weak coupling)

D ⇠ 1 g

⁴

T



^(M)

⌘ M

²

!

²

3T

^q₂

X

i

2T ⇢

ⁱⁱ_V

(!)

!

⌘⌧!<!U V

!

_{U V}

Momentum diffusion coefficient



^(M)

= 2T

²

/D

For large quark mass the transport peak is very narrow even for strong coupling and its difficult to reconstruct it accurately from Euclidean correlator calculated on the lattice area under the peak ~

^q₂

heavy quark coefficient ~ width of the peak

Current-current correlators in the heavy quark limit

0 1 2

ω / m

_D -1

0 1 2 3 4

[ κ ( ω )- κ ] / [g

2

C

F

T m

D2

/ 6 π ]

Landau cuts

full result - 2 (ω/m_D)² full result

 = 1 3T

X

3

i=1

!

lim

!0



M

lim

!1

M

²

q 2

Z

₁

1

dt e

^{i!(t t0)}

Z

d

³

~ x

⌧ 1 2

⇢ d ˆ J

ⁱ

(t, ~ x)

dt , d ˆ J

ⁱ

(t

⁰

, ~ 0) dt

⁰

G

_E

(⌧ ) = 1 3

^q₂

T

X

i

Z

d

³

x Dh

†

gE

_i

✓

^†

gE

_i

✓ i

(⌧, ~ x) h

†

gE

_i

✓

^†

gE

_i

✓ i

(0, ~ 0) E d ˆ J

ⁱ

dt = 1 M

n ˆ

^†

gE

ⁱ

ˆ ✓ ˆ

^†

gE

ⁱ

✓ ˆ o

+ O ⇣ 1 M

²

⌘

Caron-Huot, Laine, Moore, JHEP 0904 (2009) 053

t ! i⌧

G

_E

(⌧ ) =

Z

₁

0

d!

⇡ ⇢(!) cosh ⌧

_2T¹

! sinh

_2T^!

G

_E

(⌧ ) = 1

3 X

3

i=1

D ReTr h

U ( , ⌧ ) gE

_i

(⌧, ~ 0) U (⌧ , 0) gE

_i

(0, ~ 0) iE D ReTr[U ( , 0)] E

Integrate out , ✓

Transport coefficient ~ intercept of the spectral function not its width

 = lim

!!0

2T

! ⇢(!)

Calculating the electric field strength correlator on the lattice

Straightforward to discretize by deforming the path of the Wilson lines to spatial direction

Challenge : MC noise

multilevel algorithm + link integration (only works for pure glue theory)

Luscher, Weisz, JHEP 0109 (2010), 010; Parisi, Petronzio, Rapuano, PLB 128 (1983) 418

100 150 200

configuration -40

-20 0 20 40

a4 G lat

standard link multi multi + link

T = 2.25 T_c, β = 7.457, 64³x 24 τ T = 0.5

Francis, Kaczmarek, Laine, et al, arXiv:1109.3941, arXiv:1311.3759, PRD 92 (2015) 116003 Gnorm(⌧) = g²CF ⇡²T⁴

cos²(⇡⌧T)

sin⁴(⇡⌧T) + 1 3 sin²(⇡⌧T)

0 1 2 3 4 5

κ / T

³

1αa 1αb 1βa 1βb 2αa 2αb 2βa 2βb

3a BGM

model

strategy (i) strategy (ii) T

~

^{1.5 T}_c

10

^-1

10

⁰

10

¹

10

²

ω / T 10

^-1

10

⁰

10

¹

10

²

10

³

ρ

E

/ ( ω T

2

)

φ_UV^(a)

model 1αa model 2αa model 3a BGM

T

~

^{1.5 T}_c^{, T}_c ^{= 1.24 Λ}_MS

_

Extracting the spectral function and the diffusion constant

 = lim

!!0

2T

! ⇢(!) = (1.8 3.4)

Francis, Kaczmarek, Laine, et al, PRD 92 (2015) 116003

Fit the lattice using a forms of the spectral function constrained by low and high energy asymptotic behavior + corrections

⇢

^high

(!) = g

²

(µ

_!

)C

_F

6⇡ !

³

, µ

_!

= max(!, ⇡T )

⇢

^low

(!) = !

2T

T 3

<latexit sha1_base64="1IjxI4w4XI8sp6eYIv+fbgzfeJo=">AAAB6nicdVBNS8NAEJ34WetX1aOXxSJ4Kkka2norePFYsV/QxrLZbtulm03Y3Qgl9Cd48aCIV3+RN/+Nm7aCij4YeLw3w8y8IOZMadv+sNbWNza3tnM7+d29/YPDwtFxW0WJJLRFIh7JboAV5UzQlmaa024sKQ4DTjvB9CrzO/dUKhaJpp7F1A/xWLARI1gb6bZ5Vx4UinbpslZxvQqyS7ZddVwnI27VK3vIMUqGIqzQGBTe+8OIJCEVmnCsVM+xY+2nWGpGOJ3n+4miMSZTPKY9QwUOqfLTxalzdG6UIRpF0pTQaKF+n0hxqNQsDExniPVE/fYy8S+vl+hRzU+ZiBNNBVkuGiUc6Qhlf6Mhk5RoPjMEE8nMrYhMsMREm3TyJoSvT9H/pO2WHMNvvGLdXcWRg1M4gwtwoAp1uIYGtIDAGB7gCZ4tbj1aL9brsnXNWs2cwA9Yb58xnI2v</latexit><latexit sha1_base64="1IjxI4w4XI8sp6eYIv+fbgzfeJo=">AAAB6nicdVBNS8NAEJ34WetX1aOXxSJ4Kkka2norePFYsV/QxrLZbtulm03Y3Qgl9Cd48aCIV3+RN/+Nm7aCij4YeLw3w8y8IOZMadv+sNbWNza3tnM7+d29/YPDwtFxW0WJJLRFIh7JboAV5UzQlmaa024sKQ4DTjvB9CrzO/dUKhaJpp7F1A/xWLARI1gb6bZ5Vx4UinbpslZxvQqyS7ZddVwnI27VK3vIMUqGIqzQGBTe+8OIJCEVmnCsVM+xY+2nWGpGOJ3n+4miMSZTPKY9QwUOqfLTxalzdG6UIRpF0pTQaKF+n0hxqNQsDExniPVE/fYy8S+vl+hRzU+ZiBNNBVkuGiUc6Qhlf6Mhk5RoPjMEE8nMrYhMsMREm3TyJoSvT9H/pO2WHMNvvGLdXcWRg1M4gwtwoAp1uIYGtIDAGB7gCZ4tbj1aL9brsnXNWs2cwA9Yb58xnI2v</latexit><latexit sha1_base64="1IjxI4w4XI8sp6eYIv+fbgzfeJo=">AAAB6nicdVBNS8NAEJ34WetX1aOXxSJ4Kkka2norePFYsV/QxrLZbtulm03Y3Qgl9Cd48aCIV3+RN/+Nm7aCij4YeLw3w8y8IOZMadv+sNbWNza3tnM7+d29/YPDwtFxW0WJJLRFIh7JboAV5UzQlmaa024sKQ4DTjvB9CrzO/dUKhaJpp7F1A/xWLARI1gb6bZ5Vx4UinbpslZxvQqyS7ZddVwnI27VK3vIMUqGIqzQGBTe+8OIJCEVmnCsVM+xY+2nWGpGOJ3n+4miMSZTPKY9QwUOqfLTxalzdG6UIRpF0pTQaKF+n0hxqNQsDExniPVE/fYy8S+vl+hRzU+ZiBNNBVkuGiUc6Qhlf6Mhk5RoPjMEE8nMrYhMsMREm3TyJoSvT9H/pO2WHMNvvGLdXcWRg1M4gwtwoAp1uIYGtIDAGB7gCZ4tbj1aL9brsnXNWs2cwA9Yb58xnI2v</latexit><latexit sha1_base64="1IjxI4w4XI8sp6eYIv+fbgzfeJo=">AAAB6nicdVBNS8NAEJ34WetX1aOXxSJ4Kkka2norePFYsV/QxrLZbtulm03Y3Qgl9Cd48aCIV3+RN/+Nm7aCij4YeLw3w8y8IOZMadv+sNbWNza3tnM7+d29/YPDwtFxW0WJJLRFIh7JboAV5UzQlmaa024sKQ4DTjvB9CrzO/dUKhaJpp7F1A/xWLARI1gb6bZ5Vx4UinbpslZxvQqyS7ZddVwnI27VK3vIMUqGIqzQGBTe+8OIJCEVmnCsVM+xY+2nWGpGOJ3n+4miMSZTPKY9QwUOqfLTxalzdG6UIRpF0pTQaKF+n0hxqNQsDExniPVE/fYy8S+vl+hRzU+ZiBNNBVkuGiUc6Qhlf6Mhk5RoPjMEE8nMrYhMsMREm3TyJoSvT9H/pO2WHMNvvGLdXcWRg1M4gwtwoAp1uIYGtIDAGB7gCZ4tbj1aL9brsnXNWs2cwA9Yb58xnI2v</latexit>

Lattice calculations of the vector spectral functions:

Ding et al, PRD 83 (11) 034504

Fit parameters : Different choices of :