arXiv:hep-lat/0511028v1 15 Nov 2005
Mesons at high temperature in Nf= 2 QCD∗
Gert Aartsa, Chris Alltona, Richie Morrinb, Alan´O Caisb, Mehmet Bu˘ grahan Oktayb, Mike Peardonb,
aDepartment of Physics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, Wales, UK
bSchool of Mathematics, Trinity College, Dublin 2, Ireland
We report first results for spectral functions of charmonium in 2-flavour QCD. The spectral functions are
determined from vector and pseudoscalar correlators on a dynamical, anisotropic lattice. J/ψ and ηc are found to
survive well into the deconfined phase before melting away at T ? 2Tc. Current systematic uncertainties prevent
us from drawing any definite conclusions at this stage.
The properties of hadrons or hadronic reso-
nances above the deconfinement transition is a
subject at the heart of the current experimen-
tal programme at RHIC, where hadronic signals
are used to obtain information about the state of
matter inside the fireball. The questions of in-
terest include the issue of which hadrons survive
as bound states in the quark–gluon plasma, and
up to which temperature; as well as the trans-
port properties of light and heavy quarks in the
These properties are all encoded in the spec-
tral functions ρ(ω,? p), which are related to the
imaginary-time correlator GΓ(τ,? p) according to
GΓ(τ,? p) =
ρΓ(ω,? p)K(τ,ω)dω, (1)
where the subscript Γ correspond to the different
quantum numbers. The kernel K is given by
K(τ,ω) =cosh[ω(τ − 1/2T)]
= eωτnB(ω) + e−ωτ[1 + nB(ω)].
and nB is the Bose–Einstein distribution func-
The spectral function can be extracted from
the lattice correlators G(τ) using the Maximum
∗Talk by JIS at Workshop on Computational Hadron
Physics, University of Cyprus, 14–17 September 2005.
Entropy Method (MEM) . For this to work
and give reliable results, it is necessary to have a
sufficient number of points in the euclidean time
direction. This will be prohibitively expensive,
especially when dynamical quarks are included,
unless anisotropic lattices are used, with a tempo-
ral lattice spacing much smaller than the spatial
We will here focus on the charmonium S-wave
states ηcand J/ψ at zero momentum, which have
attracted much attention following the suggestion
 that J/ψ suppression could be a probe of de-
confinement. Potential model calculations using
the heavy quark free energy have tended to sup-
port this picture. However, previous simulations
in the quenched approximation [3,4,5] indicate
that contrary to this, J/ψ may survive up to tem-
peratures as high as 1.5−2Tc. Recently, potential
model calculations using the internal energy of
the heavy-quark pair have reached the same con-
clusion, and using the most recent lattice data
 these models indicate a qualitatively similar
picture in the case of Nf= 2 QCD [6,7].
In this study we attempt to determine directly
the spectral functions of charmonium in 2-flavour
QCD using anisotropic lattices and the Maximum
2. SIMULATION DETAILS
We use the Two-plaquette Symanzik Improved
gauge action  and the fine-Wilson, coarse-
Hamber-Wu fermion action  with stout-link
smearing . The process of tuning the ac-
tion parameters, and the parameters used, are de-
scribed in more detail in [11,12]. The parameters
correspond to a spatial lattice spacing as≈ 0.2fm
with an anisotropy ξ = as/at≈ 6. The sea quark
mass corresponds to mπ/mρ≈ 0.55.
anisotropies, and a quenched study  found
that no mass-dependent tuning of the quark
anisotropy was necessary up to valence quark
masses well beyond charm.
longer to be the case for dynamical quarks; in-
deed, for the parameters used here the anisotropy
determined from the pion dispersion relation was
found to be ξℓ
q∼ 6.4, in rough agreement with
the gluon anisotropy, while the anisotropy from
the charmonium dispersion relation was found to
q∼ 8. This issue is still under investigation.
For this preliminary study we have used a spa-
tial volume of N3
urations, sampled every 10 HMC trajectories, for
Nt= 48,32,24 and 16. Nt= 32 corresponds to
a temperature close to the pseudocritical temper-
ature Tc, although larger lattices will be needed
to determine this with any precision. We have
computed charmonium correlators in the pseu-
doscalar (ηc) and vector (J/ψ) channels with bare
charm quark mass atmc= 0.1. The charmonium
spectrum in the hadronic phase is presented in
[12,13].In this study we have used local (un-
This appears no
s= 83and generated 100 config-
? x,? y,t
Γ(? x,t)MΓ(? y,t + τ)?, (3)
MΓ(? x,t) =¯ψ(? x,t)Γψ(? x,t),Γ = γ5,γi. (4)
All-to-all propagators  have been used to im-
prove the signal and sample information from the
entire lattice. The propagators were constructed
with no eigenvectors and two noise vectors diluted
in time, colour and even/odd in space.
The MEM analysis has been performed with
the continuum free spectral function ω2as default
model, using the euclidean correlators in a time
window starting at τ = 2, and cutting off the
energy integral in (1) at atωmax= 6.
Figure 1. Average Polyakov loop as a function of
In an attempt to locate the pseudocritical tem-
perature on these lattices, we performed simula-
tions varying Nt in the range 28–40, measuring
the real part of the Polyakov loop ?L? and its
susceptibility. The results for the Polyakov loop
are shown in fig. 1.As can be seen from the
figure, the Polyakov line in the transition region
follows a linear behaviour in temperature, and
therefore no pseudocritical temperature can be
determined on these lattices. It will be necessary
to use larger lattices to determine Tc. For this
reason, and because of the uncertainties regard-
ing the anisotropy, we will refrain from quoting
results in terms of Tc.
The free-fermion spectral functions are shown
in fig. 2, for Nt = 32 and 24, and in the con-
tinuum. The most striking feature of the lattice
free spectral functions is the spike at atω ≈ 0.6.
This means that lattice artefacts are big at this
point, and results for spectral functions cannot
be trusted in this region. The lattice functions
undershooting the continuum curve appears to
be a mass-dependent, O(atm) or O(a2
fect, evidenced by the effect being much larger
at atm = 0.2 .
The spectral functions obtained from the MEM
Figure 2. Free-fermion spectral functions in the
pseudoscalar and vector channels, for bare quark
mass atm0= 0.1.
Figure 3. Pseudoscalar (ηc) spectral function for
analysis are shown in figs. 3 and 4 for ηcand J/ψ
respectively. The main peak position for Nt= 48
agrees within errors with the mass obtained for
the respective particles on the same lattices, using
a variational basis of smeared operators [12,13].
The second peak coincides with the cusp in the
free spectral function in fig. 2, and it can there-
fore be concluded that this is primarily a lattice
artefact. The radially excited states cannot be
resolved with our present statistics.
The results in figs. 3 and 4 indicate that the 1S
states survive in the medium up to well beyond
Figure 4. Vector (J/ψ) spectral function for dif-
the deconfinement temperature, finally melting
away at T ? 2Tc. This is in qualitative agree-
ment with recent potential model results using
the static quark–antiquark internal energy .
Using the colour singlet internal energy U1yields
a dissociation temperature of Tdis ∼ 2Tc, while
a potential constructed to exclude the gluon in-
ternal energy  yields Tdis∼ 1.4Tc. Given
the uncertainties in this calculation, both these
results are consistent with the present data.
It is not clear whether the apparent stronger
binding in ηc for Nt = 24 is significant, in par-
ticular given that no such effect is observed for
J/ψ. Higher statistics will be needed to resolve
this. If it is confirmed, it might be in line with
potential model calculations using U1, which is
strongly peaked around Tc.
Using an anisotropic lattice, we have performed
the first calculation of charmonium spectral func-
tions in 2-flavour QCD. The results appear to
confirm the picture emerging from quenched sim-
ulations, that J/ψ and ηcsurvive until well into
the deconfined phase.
These simulations have been performed with
parameters that are not fully tuned, giving rise to
significant systematic uncertainties. A particular
issue is the tuning of the charm quark anisotropy.
We are in the process of obtaining a fully tuned
4 Download full-text
parameter set, and simulations at these parame-
ters will be carried out in the very near future.
Firm conclusions will have to await these simula-
The small lattice volume — only (1.6fm)3—
is also a major source of systematic uncertainty,
in particular as it has prevented a determination
of the pseudocritical temperature. Future simula-
tions will be carried out on larger spatial volumes.
It will also be important to increase the statistics,
in order to resolve excited states and disentangle
the effects of thermal width and finite statistics.
We will also be carrying out a systematic study
of the effects of using different default models, in-
cluding the free lattice spectral functions, and dif-
ferent time and energy ranges in the MEM anal-
ysis. Initial indications are that our results are
relatively robust against such changes.
The coarse spatial lattice is an issue in that it
gives rise to significant, mass-dependent, lattice
artefacts as shown in fig. 2. It will ultimately be
necessary to repeat the calculation on a finer lat-
tice; this will however require a new nonperturba-
tive tuning and is therefore not on the immediate
horizon. Some information about lattice spac-
ing effects may be gleaned from quenched stud-
ies using the same action, which are considerably
cheaper as the quark and gluon anisotropies can
be tuned independently.
We are planning to compute the spectral func-
tions also at non-zero momentum , which con-
tain additional information not found at zero mo-
mentum, and which may relate more directly to
experimental data taken at non-zero momentum.
We will also study light vector meson corre-
lators at zero and non-zero momentum, which
can be related to dilepton production rates in the
plasma. Finite lattice spacing effects are expected
to be less of a problem in this case, so the current
lattices are likely to be suitable for this purpose.
This will proceed once the fully tuned parameter
set is ready.
We thank the organisers of Workshop on Com-
putational Hadron Physics for a pleasant and in-
teresting workshop.This work has been sup-
ported by the IRCSET Embark Initiative award
SC/03/393Y, SFI grant 04/BRG/P0275 and the
IITAC PRTLI initiative.
Jimmy Juge, Sin´ ead Ryan and Simon Hands for
stimulating and fruitful discussions.
We wish to thank
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