Across the deconfinement transition:
parity doubling in the nucleon sector&diffusion of light quarks
Gert Aarts

Kyoto, March 2015 – p. 1

Introduction
QCD phase diagram:at T 6= 0 and µ = 0
from hadronic to quark-gluon plasma
standard observables:pressure, entropy, fluctuations
confinement, chiral symmetry
light mesons, quarkonia...
less standard observables:transport
baryons...
Kyoto, March 2015 – p. 2

Outlineacross the deconfinement transition:
nucleons: medium effects and parity doubling

conductivity and charge diffusion
all results:
anisotropic Nf = 2 + 1 ensembles with Wilson-clover
fermions
part of FASTSUM collaboration

Kyoto, March 2015 – p. 3

nucleons across the deconfinementtransition
Benjamin Jäger, GA, Chris Allton, Simon Hands, Kristi Praki andJonivar Skullerud
arXiv:1502.03603 [hep-lat]

Kyoto, March 2015 – p. 4

Mesons/baryons in a medium
mesons in a medium very well studied
thermal broadening and mass shift in hadronic phasedeconfinement/melting in the QGP
quarkonia survival as thermometer
conductivity/dileptons from vector current
chiral symmetry restoration
relatively easy on the latticehigh-precision correlators
what about baryons?

Kyoto, March 2015 – p. 5

Baryons in a medium
lattice studies of baryons at finite temperature very limited
screening masses De Tar and Kogut 1987... with a small chemical potential QCD-TARO: Pushkina, deForcrand, Kim, Nakamura, Stamatescu et al 2005

temporal correlators Datta, Gupta, Mathur et al 2013
not much more (afaik)
but what about
in-medium modification?
chiral symmetry?
parity doubling?
...
Kyoto, March 2015 – p. 6

Nucleons in a medium
simplest nucleon operator TON (x, τ ) = ǫabc ua (x, τ ) ub (x, τ )Cγ5 dc (x, τ )essential difference with mesons: role of parity
PON (x, τ ) = γ4 ON (−x, τ )positive/negative parity operators
ON± (x, τ ) = P± ON (x, τ )1
P± = (1 ± γ4 )2euclidean correlators
Z
G± (τ ) = d3 x ON± (x, τ )ON± (0, 0)
Kyoto, March 2015 – p. 7

Mesons/baryons in a medium
meson versus baryon correlators
meson correlators symmetric around τ = 1/2T
baryon correlators not symmetriccontain both positive and negative parity channels
for G+ (τ ):positive parity state propagates with τ
negative parity state propagates with 1/T − τminimum typically not at τ = 1/2TZ
G± (τ ) = d3 x ON± (x, τ )ON± (0, 0)
Kyoto, March 2015 – p. 8

Baryons in a medium
example: nucleon ground state
G± (τ ) = A± e−m± τ + A∓ e−m∓ (1/T −τ )nucleon:m+ = mN = 0.939 GeV
m− = mN ∗ = 1.535 GeV

no parity doubling: manifestation of chiral symmetry
breaking
parity doubling:
degeneracy between +/− parity channels
sufficient condition is unbroken chiral symmetry
G± (τ ) = G∓ (τ ) = G± (1/T − τ )Kyoto, March 2015 – p. 9

On the lattice
FASTSUM ensembles
Nf = 2 + 1 dynamical quark flavours, Wilson-clover

many temperatures, below and above Tc
anisotropic lattice, as /aτ = 3.5, many time slices
strange quark: physical value
two light flavours: somewhat heavy mπ = 384(4) MeV
Ns

24

32

24

24

32/24

32/24

32/24

24

32/24

Nτ

128

48

40

36

32

28

24

20

16T /Tc

0.24

0.63

0.76

0.840.95

1.09

1.27

1.52

1.90

Ncfg

400

600

500

500

500

500

500

1000

1000

tuning and Nτ = 128 data from HadSpec collaboration

Kyoto, March 2015 – p. 10

Nucleons in a medium
various interpolation operators, here simplest one
 T
ON (x, τ ) = ǫabc ua (x, τ ) ub (x, τ )Cγ5 dc (x, τ )Gaussian smearing for multiple sources and sinks
same smearing parameters at all temperatures
questions:in-medium effects below Tc
parity doubling
deconfinement transition, chiral symmetry

Kyoto, March 2015 – p. 11

Lattice correlators
euclidean correlator G+ (τ )010

T/Tc=1.90
T/Tc=1.52
T/Tc=1.27
T/Tc=1.09

T/Tc=0.95
T/Tc=0.84
T/Tc=0.76-1G(τ)/G(0)

10-2

10-3

10 0

0.2

0.4

τT0.6

0.8

1

not symmetric around τ = 1/2T below Tc
more symmetric as temperature increases
Kyoto, March 2015 – p. 12

Nucleons in a medium
separate positive and negative parity channels
010

positive parity

negative parity-1

G(τ)/G(0)10

T/Tc=1.90
T/Tc=1.52
T/Tc=1.27
T/Tc=1.09
T/Tc=0.95
T/Tc=0.84
T/Tc=0.76
T/Tc=0.24-210-310-410-510 081624

08162432

τ/aτbelow Tc : m− > m+m+ = mN , m− = mN ∗much more T dependence in negative-parity channel
Kyoto, March 2015 – p. 13

Nucleons in a medium
exponential fits/effective masses below Tc
T /Tcaτ mNaτ mN ∗mN [GeV]mN ∗ [GeV]

0.24
0.76
0.84
0.95

0.213(5)0.209(16)0.192(17)0.198(25)0.33(5)0.28(3)0.28(2)0.22(4)1.20(3)
1.18(9)1.08(9)1.12(14)1.9(3)
1.6(2)
1.6(1)
1.3(2)

m± larger than in Nature (probably ∼ heavy pions)mass splitting mN ∗ − mN ∼ 700 MeV
nucleon ground state largely T independentN ∗ ground state significant temperature dependence

relevant for heavy-ion phenomenology?
Kyoto, March 2015 – p. 14

Nucleons in a medium
parity doublingcorrelator ratio
G+ (τ ) − G+ (1/T − τ )R(τ ) =
G+ (τ ) + G+ (1/T − τ )ifno parity doubling and m− ≫ m+ : R(τ ) = 1
parity doubling: R(τ ) = 0note
R(1/T − τ ) = −R(τ ) and R(1/2T ) = 0Kyoto, March 2015 – p. 15

Parity doubling
1.25

T/Tc=0.24
T/Tc=0.76
T/Tc=0.84
T/Tc=0.95
T/Tc=1.09
T/Tc=1.27
T/Tc=1.52
T/Tc=1.90

1R(τ)

0.75

0.5

0.25

00

0.1

0.2

τT

0.3

0.4

0.5

ratio close to 1 below Tc , decreasing uniformly
ratio close to 0 above Tc , parity doubling
technical note: smearing essential
Kyoto, March 2015 – p. 16

Quasi-order parameter
integrated ratio

R=P 12 Nτ −1n=1R(τn )/σ 2 (τn )P 12 Nτ −1n=111/σ 2 (τn )

0.8

R

0.6

0.4

0.2

000.5

11.5

2

T/Tc

crossover behaviour, tied with deconfinement transition
Kyoto, March 2015 – p. 17

Quasi-order parametersignal depends quantitatively on interpolating operator
different (more complicated) operator
more suppression of excited states
T/Tc=0.24
T/Tc=0.76
T/Tc=0.85
T/Tc=0.95
T/Tc=1.09
T/Tc=1.27
T/Tc=1.52
T/Tc=1.90

1R(τ)

0.75

0.5

10.8

0.6

R

1.25

0.4

0.25

0.2000

0.1

0.2

τT

0.3

0.4

0.5

00.5

1

1.5

2

T/Tc

but semi-quantitative agreement
parity doubling coincides with deconfinementtransition: tied to restoration of chiral symmetry
Kyoto, March 2015 – p. 18

Summary: nucleons in medium
N mostly temperature independent below Tc

significant T dependence in N ∗ channel
reduction in massparity doubling above Tc
closely linked to deconfinement transition and chiralsymmetry restoration
outlookbaryons with strange quarks
role of smearing
use chiral fermions

Kyoto, March 2015 – p. 19

diffusion and conductivity
Alessandro Amato, Pietro Giudice, GA, Chris Allton, Simon Hands andJonivar Skullerud
arXiv:1307.6763 [hep-lat] (PRL), arXiv:1412.6411 [hep-lat] (JHEP)

Kyoto, March 2015 – p. 20

Transport coefficients
dynamics on long length and timescales:effective theory: hydrodynamics
ideal hydrodynamics: equation of stateviscous hydro: transport coefficientsshear/bulk viscosity, conductivity, . . .
depend on underlying microscopic theory
typically:large in weakly interacting theory
small in strongly coupled systems
perfect-fluid paradigm: η/s = 1/4π (holography)
Kyoto, March 2015 – p. 21

Transport coefficientsin the pastemphasis on viscosity
bound from holography η/s = 1/4π
scale invariance and bulk viscosity
recently
more interest in electrical conductivity
role in heavy-ion collisions
charge density fluctuations
strong magnetic fields...
Kyoto, March 2015 – p. 22

Conductivity/diffusion
electrical conductivity σcharge susceptibility χboth σ and χ proportional to EM factor
Cem = e2

X

qf2

f2 1
qf = , −3 3diffusion coefficient D = σ/χ
Cem cancels

finite large Nc limit
weak coupling: D ∼ 1/g 4 Tstrong coupling: D = 1/2πT (holography)

Kyoto, March 2015 – p. 23

Diffusion coefficient
new result: D across the deconfinement transition
2.5

0.75

1.00

T /Tc
1.25
1.50

1.75

2.00

2πT D

2.0
1.5
1.0
0.5
0.0
100

150

200

250
T [MeV]

300

350

400

minimal around Tc
order of magnitude D ∼ 1/2πT

Kyoto, March 2015 – p. 24

Conductivity/diffusion
linear response: Kubo relation
1σ = lim
ρii (ω, 0)
ω→0 6ω

whereρµν (x) = h[jµ (x), jν (0)]ieq

is current-current spectral function, jµ is EM current
real-time correlator in equilibriumon the lattice: euclidean correlator
related to spectral function
Zcosh[ω(τ − 1/2T )]G(τ ) = dω K(τ, ω)ρ(ω)
K(τ, ω) =sinh(ω/2T )inversion/analytical continuation
Kyoto, March 2015 – p. 25

Conductivity from the latticeuse same ensembles
T /Tc = 0.24, 0.76, 0.84, 0.95, 1.09, 1.27, 1.52, 1.90
Nf = 2 + 1 dynamical quark flavours

conserved lattice current (no renormalisation required)
2e u e d e semjµ = jµ − jµ − jµ333

strange and up/down quarks: mass effect incorrelators
spectral functions
flavour susceptibilities
conductivity
Kyoto, March 2015 – p. 26

Conserved current-current correlator
ratio with free massless lattice correlator
1.2

mq = mu,dmq = ms

T /Tc

T /Tc

G(τ )/Gfree
LAT (τ )

1.0
0.8
0.6
0.4
0.2
0.0

1.90
1.52

0.1

1.27
1.09

0.2

0.3

0.95
0.84

0.4

0.5

0.0

0.1

τT

0.76
0.63

0.2

0.3

0.4

0.5

τT

distinction between below/above Tc
quark mass dependence
Kyoto, March 2015 – p. 27

Conserved current-current correlator
finite-size effects?
GNs =32/GNs =24
1.04
1.02
1.00
0.98
0.96

T /Tc = 1.90

1.04
1.02
1.00
0.98
0.96

T /Tc = 1.27

1.04
1.02
1.00
0.98
0.96
1.04
1.02
1.00
0.98
0.96
0.0

T /Tc = 1.09

T /Tc = 0.95

0.1

0.2

0.3

0.4

0.5
τT

0.6

0.7

0.8

0.9

1.0

only change is in spatial volume, Ls = 2.9 and 3.9 fm
no finite-size effectsKyoto, March 2015 – p. 27

Spectral functions
from correlator to spectral function
Zcosh[ω(τ − 1/2T )]G(τ ) = dω K(τ, ω)ρ(ω)
K(τ, ω) =sinh(ω/2T )inversion/analytical continuation
use Maximal Entropy Method (MEM)
Asakawa, Hatsuda & Nakahara 2001

with 1/ω instability fixed
GA, Allton, Foley, Hands & Kim 2007

many systematic checks (see below)

Kyoto, March 2015 – p. 28

Spectral functions
ρ(ω)/ω 2
mq = mu,d1.4

1.90Tc
1.09Tc
0.63Tc1.2

ρ(ω)/ω 2

1.0

1.0
0.8

0.60.60.4

0.4

0.2

0.2

0.00.01234
ω [GeV]51.90Tc
1.09Tc
0.63Tc1.20.80mq = ms

1.4

601234
ω [GeV]

56peak below Tc corresponds to ρ, φ particle
divergence as ω → 0 corresponds to transport peak
Kyoto, March 2015 – p. 29

Spectral functions
ρ(ω)/ωT87

ρ(ω)/ωT

6mq = mu,d1.90Tc
1.09Tc
0.63Tc543mq = ms

12108

1.90Tc
1.09Tc
0.63Tc642
10
0.0 0.2 0.4 0.6 0.8 1.0 1.2ω [GeV]

200.0 0.2 0.4 0.6 0.8 1.0 1.2
ω [GeV]

intercept: conductivity
temperature and mass dependent
Kyoto, March 2015 – p. 29

Conductivity−1 σ/T
conductivity Cem

0.40

0.50

0.75

1.00

T /Tc
1.25

1.50

1.75

2.00

0.40

light+strange

0.35

0.25

0.25
−1σ/T
Cem

0.30

0.20
0.15

0.05

0.05

0.00

0.00
200
250
T [MeV]

300

350

1.50

1.75

2.00

0.15
0.10

150

1.00

0.20

0.10

100

0.75
light
strange

0.35

0.30

−0.05

0.50

T /Tc
1.25

−0.05

100

150

200
250
T [MeV]

300350temperature and mass dependent
agreement with previous results above Tc
Kyoto, March 2015 – p. 30

Conductivity−1 σ/T
conductivity Cem

agreement with previous results above Tc
Kyoto, March 2015 – p. 31

Susceptibilies
fluctuations of isospin, electrical charge, baryon
number, flavour
1.0T /Tc
T /Tc
0.75 1.00 1.25 1.50 1.75 2.00

0.75 1.00 1.25 1.50 1.75 2.00

0.8

χ/χSB

0.6
0.4
0.2
0.0

χIχQ
χB

χllχss−0.2
100 150 200 250 300 350 400 100 150 200 250 300 350 400
T [MeV]T [MeV]agreement with previous (mostly staggered) results
some flavour dependence

Kyoto, March 2015 – p. 32

Diffusion coefficient
combination of results: D = σ/χQ
2.5

0.75

1.00

T /Tc
1.25
1.50

1.75

2.00

2πT D

2.0
1.5
1.0
0.5
0.0
100

150

200

250
T [MeV]

300

350

400

consistent with strongly coupled plasma

Kyoto, March 2015 – p. 33

Systematic checks
results stable against variations in MEM?
default model
euclidean time range
discretisation...
some illustrations

Kyoto, March 2015 – p. 34

MEM stability
anisotropic lattice, as /aτ = 3.5
use all or 1 in 2 or 1 in 30.5
0.4

all1 in 2
1 in 3−1
Cem
σ/T

0.3
0.2
0.1
0.0−0.1

40363228Nτ

2420

16

stable at larger Nτ – anisotropy beneficial at smaller Nτ
Kyoto, March 2015 – p. 35

MEM stabilityG(τ )10−2−1
σ/T
Cem

0.40
0.35
0.30

1.09 Tc
1.27 Tc
1.52 Tc

0.25
100.20−30.15nτ = 610−4468 10 12 14 16 18 20

0.10
0.05
0.00302826

242220180.3010−20.25

0.95 Tc
1.09 Tc
1.27 Tc

0.20compare withtime range

0.15

10−30.10

10−4

use restricted time range

4nτ = 80.05

60.00
34

8 10 12 14 16 18 203230282624maximal

220.3010−20.25

robust signal0.84 Tc
0.95 Tc
1.09 Tc

0.20
0.15

10−3

0.10

10−4

4

nτ = 100.05

60.00
38

8 10 12 14 16 18 20
τ

3634

32
Nτ

30

2826

Kyoto, March 2015 – p. 35

Summary
conductivity/diffusion:
computed across deconfinement transition
D minimum around Tc , D ∼ 1/2πT

consistent with strongly coupled plasma
quark mass dependencenucleons:N largely T independent below Tc
N ∗ substantial T dependence

parity doubling above Tc

Kyoto, March 2015 – p. 36