M ESON SPECTRAL FUNCTIONS AT NON - ZERO
MOMENTUM IN HIGH - TEMPERATURE QCD
Gert Aarts, Chris Allton, Justin Foley, Simon Hands
Physics Department, University of Wales Swansea
Seyong Kim
Sejong University, Seoul
UKQCD 23/01/06 – p.1
ECT* D OCTORAL T RAINING P ROGRAMME 2006
ADVERTISEMENT ON BEHALF OF
S IMON H ANDS
Computational Techniques in Strongly Interacting Systems
Trento March 13 - May 26
daily lectures from recognised experts on
lattice QCD
QCD thermodynamics
numerical algorithms for fermions
role of topological excitations
non-equilibrium quantum field theory
quantum MC in many-body physics
website: http://www.ect.it/
enquiries: Simon Hands s.hands@swan.ac.uk
deadline for applications: February 3, 2006
UKQCD 23/01/06 – p.2
H OT QCD, QUARK - GLUON PLASMA
MOTIVATION
deconfined phase: quarks and gluons
different dynamics ⇒ different interests, observables
quark-gluon plasma at Tc . T . 3Tc :
survival of bound states: charmonium, ...
dilepton and photon production: Πµν ∼ hj µ j ν i
transport coefficients: viscosity, conductivity, ...
very relevant for heavy-ion phenomenology
observables best expressed in terms of spectral functions
example: differential rate
for dilepton production
(non-perturbative in αstrong )
αem
dΓ
µ
0
=
nB (k )ImΠRµ (K)
4
4
2
d K
12π K
UKQCD 23/01/06 – p.3
S PECTRAL FUNCTIONS
REAL VERSUS IMAGINARY TIME
relation between euclidean correlator and spectral function:
Z ∞
dω
GH (τ, p) =
K(τ, ω)ρH (ω, p)
2π
0
cosh[ω(τ − 1/2T )]
with the kernel K(τ, ω) =
sinh(ω/2T )
Maximal entropy method Asakawa & Hatsuda (2000) (revived)
potential difficulty:
for small ω , kernel K ∼ 2T /ω , independent of τ
GH (τ ) insensitive to details of ρH (ω) at soft energies
soft energies interesting for high T QGP
first try: meson correlators at non-zero momentum
UKQCD 23/01/06 – p.4
M ESON SPECTRAL FUNCTIONS
AT NON - ZERO MOMENTUM
meson spectral functions:
†
ρH (t, x) = h[JH (t, x), JH
(0, 0)]i
euclidean correlators:
†
GH (τ, x) = hJH (τ, x)JH
(0, 0)i
currents:
JH (τ, x) = q̄(τ, x)ΓH q(τ, x)
with
ΓH = {11, γ5 , γ µ , γ µ γ5 }
expectation:
below Tc : mesons moving in the heatbath
simple p
sharply peaked quasiparticle spectral functions
at ω = p2 + m2 (+ subdominant continuum)
above Tc : ‘free’ quarks, only continuum contribution
study analytically at infinite temperature (one loop)
UKQCD 23/01/06 – p.5
S CALAR AND PSEUDOSCALAR
large ω : ρPS,S (ω, p) ∼ ω 2 , dimensional analysis
small ω : ρPS,S (ω, p) ∼ ω , odd
show ρPS,S /ω 2 :
staggered
ρ(ω,p)/ω
2
Wilson
PS, Nτ=16
S, Nτ=16
PS, cont
S, cont
p/T=4, m/T=1
0.6
0.4
0.2
0
0
20
40
60 0
ω/T
20
40
60
ω m: scalar and pseudoscalar degenerate for
continuum and staggered, broken for Wilson fermions
lightcone
non-trivial low energy region
UKQCD 23/01/06 – p.6
S CALAR AND PSEUDOSCALAR
large ω : ρPS,S (ω, p) ∼ ω 2 , dimensional analysis
small ω : ρPS,S (ω, p) ∼ ω , odd
show ρPS,S /T 2 :
staggered
ρ(ω,p)/T
2
Wilson
p/T=4, m/T=1
0.6
PS, cont
S, cont
PS, Nτ=16
S, Nτ=16
0.4
0.2
0
0
2
4
0
ω/T
2
4
6
ω ∼ m: scalar and pseudoscalar non-degenerate
lightcone affected by lattice artefacts
below the lightcone: Landau damping
UKQCD 23/01/06 – p.6
Q UENCHED QCD WITH STAGGERED FERMIONS
PRELIMINARY RESULTS , IN PROGRESS
quenched QCD:
ensembles below and above Tc with Nτ = 24 fixed
(for MEM purposes)
below Tc :
483 × 24, β = 6.5, a−1 ∼ 4 GeV, T ∼ 160 MeV
above Tc :
643 × 24, β = 7.192, a−1 ∼ 10 GeV, T ∼ 420 MeV
50 configurations analyzed
staggered quarks:
ma = 0.01, 0.05, 0.125
m/T = 0.24, 1.2, 3
UKQCD 23/01/06 – p.7
Q UENCHED QCD WITH STAGGERED FERMIONS
PRELIMINARY RESULTS , IN PROGRESS
quenched QCD:
ensembles below and above Tc with Nτ = 24 fixed
(for MEM purposes)
below Tc :
483 × 24, β = 6.5, a−1 ∼ 4 GeV, T ∼ 160 MeV
above Tc :
643 × 24, β = 7.192, a−1 ∼ 10 GeV, T ∼ 420 MeV
50 configurations analyzed
staggered quarks:
ma = 0.01, 0.05, 0.125
m/T = 0.24, 1.2, 3
chiral symmetry breaking/restoration:
pseudoscalar vs. scalar for light quarks
UKQCD 23/01/06 – p.7
Q UENCHED QCD WITH STAGGERED FERMIONS
PRELIMINARY RESULTS , IN PROGRESS
cold, T ≈ 160 MeV: confined phase
10
π m/T=0.24
σ m/T=0.24
π m/T=1.2
σ m/T=1.2
π m/T=3
σ m/T=3
1
3
48 x24
β=6.5
0.01
3
a G(τ,0)
0.1
0.001
0.0001
1e-05
0
4
8
12
nτ
16
20
24
UKQCD 23/01/06 – p.8
Q UENCHED QCD WITH STAGGERED FERMIONS
PRELIMINARY RESULTS , IN PROGRESS
hot, T ≈ 420 MeV: chiral symmetry restoration
10
π m/T=0.24
σ m/T=0.24
π m/T=1.2
σ m/T=1.2
π m/T=3
σ m/T=3
1
3
64 x24
β=7.192
0.01
3
a G(τ,0)
0.1
0.001
0.0001
1e-05
0
4
8
12
nτ
16
20
24
UKQCD 23/01/06 – p.8
N ON - ZERO MOMENTUM
WITH TWISTED BOUNDARY CONDITIONS
momenta on the lattice p~ =
2π
n
L~
hydrodynamic regime: many momenta with p . T
large volume, large ratio Nσ /Nτ , large Nτ ?
twisted boundary conditions
(long history, Flynn et.al. 2005)
quark field: ψ(xi + L) = eiθi ψ(xi )
meson: two twist angles θ~1 , θ~2
meson momentum: p~ =
2π
θ~1 −θ~2
n− L
L~
on 643 × 24: ∼ 20 momenta in the range 0 < p/T < 3.8
UKQCD 23/01/06 – p.9
H IGH TEMPERATURE QUENCHED QCD
SMALL ( BARE ) QUARK MASS ,
m/T = 0.24
hot, T ≈ 420 MeV
0.1
3
a GPS(τ,p)
1
0.01
3
64 x24
β=7.192
m/T=0.24
increasing momentum
0.001
0
4
8
12
16
20
24
nτ
pseudoscalar: ∼ 20 different momenta
UKQCD 23/01/06 – p.10
H IGH TEMPERATURE QUENCHED QCD
SMALL ( BARE ) QUARK MASS ,
m/T = 0.24
hot, T ≈ 420 MeV
3
a GPS(τ,p)
0.01
3
64 x24
β=7.192
m/T=0.24
increasing momentum
0.001
8
12
16
nτ
zoom in: small but non-zero effect, correctly ordered
UKQCD 23/01/06 – p.10
H IGH TEMPERATURE QUENCHED QCD
SMALL ( BARE ) QUARK MASS ,
m/T = 0.24
hot, T ≈ 420 MeV
1
G(τ,p)/G(τ,0)
0.95
0.9
3
64 x24
β=7.192
m/T=0.24
0.85
0
4
pseudoscalar
scalar
8
12
16
20
24
nτ
chiral symmetry restoration at non-zero momentum
UKQCD 23/01/06 – p.10
H IGH TEMPERATURE QUENCHED QCD
SMALL ( BARE ) QUARK MASS ,
m/T = 0.24
hot, T ≈ 420 MeV
1
G(τ,p)/G(τ,0)
0.95
0.9
pseudoscalar
3
64 x48
free
am=0.01
0.85
0
4
8
12
16
20
24
nτ
for comparison: free staggered quarks
UKQCD 23/01/06 – p.10
H IGH TEMPERATURE QUENCHED QCD
SMALL ( BARE ) QUARK MASS ,
m/T = 0.24
cold, T ≈ 160 MeV
1
G(τ,p)/G(τ,0)
0.9
0.8
0.7
pseudoscalar
3
48 x24
β=6.5
m/T=0.24
0.6
0.5
0
4
8
12
16
20
24
nτ
ratio in the confined phase: larger relative effect
UKQCD 23/01/06 – p.10
S TATUS
clear non-zero signal at non-zero momentum
hot lattice: relative size roughly in agreement with free
calculation at infinite temperature
standard zero-temperature analysis does not work:
deconfined plasma, no bound states
apply Maximal Entropy Method (in progress)
note: for staggered fermions spectral relation reads
Z ∞
i
h
dω
GH (τ, p) =
K(τ, ω) ρH (ω, p) − (−1)τ /aτ ρ̃H (ω, p)
2π
0
MEM: independent analysis on even/odd timeslices
Wilson fermions: in progress
UKQCD 23/01/06 – p.11
O UTLOOK :
DYNAMICAL FERMIONS
ONLY ONE STUDY, ZERO MOMENTUM
Charmonium spectral functions in Nf=2 QCD
TrinLat + SwanLat (R. Morrin, A. O’Cais, B. Oktay, M. Peardon,
J.-I. Skullerud, G. A. and C. Allton) PoS(LAT2005) 176 [hep-lat/0509115]
many time slices: anisotropic lattice, 83 × 48, 32, 24, 16
survival of charmonium to
approx. 1.5Tc , gone at 2Tc 15
N =48
in line with previous
N =32
N =24
quenched studies
N =16
t
t
t
ρ(ω)/ω
still large systematic and
statistical uncertainties
2
t
10
5
J/ψ at T /Tc ≈ 0.75, 1.1, 1.5, 2.2
0
QCDOC?
0
0.2
0.4
atω
0.6
0.8
1
UKQCD 23/01/06 – p.12