We investigate medium effects on mesonic screening lengths for QCD with
2-flavours of dynamical staggered quarks on lattices with cutoff a=1/6T.
Denoting the cross-over temperature by $T_c$, we vary the temperature T from
0.89$T_c$ to 1.92$T_c$, spanning both the hadronic and quark gluon plasma
phases. While chiral symmetry restoration in the vector channel appears to take
place near $T_c$, it is seen in the scalar channel only above 1.33 $T_c$.
Varying spatial lattice sizes, we find very little volume dependence in our
results at 0.94 $T_c$. We discuss the stability of the scalar meson at these
temperatures.Comment: Talk presented at The XXVIII International Symposium on Lattice Field
  Theory, Lattice2010, June 14-19, 2010, Villasimius, Ital

Banerjee, Debasish

Gavai, Rajiv

Gupta, Sourendu

English

arXiv

We investigate medium effects on mesonic screening lengths for QCD with 2-flavours of dynamical staggered quarks on lattices with cutoff a=⅙T. Denoting the cross-over temperature by Tc, we vary the temperature T from 0.89Tc to 1.92Tc, spanning both the hadronic and quark gluon plasma phases. While chiral symmetry restoration in the vector channel appears to take place near Tc, it is seen in the scalar channel only above 1.33 Tc. Varying spatial lattice sizes, we find very little volume dependence in our results at 0.94 Tc. We discuss the stability of the scalar meson at these temperatures

Publications of the IAS Fellows

P
oS(Lattice 2010)168
The scalar does not decay at finite temperatures
Debasish Banerjee∗, Rajiv Gavai and Sourendu Gupta
Department of Theoretical Physics, Tata Institute of Fundamental Research,
Homi Bhabha Road, Mumbai 400005, India
E-mail:
debasish@theory.tifr.res.in,gavai@tifr.res.in,sgupta@tifr.res.in
We investigate medium effects on mesonic screening lengths for QCD with 2-flavours of dynam-
ical staggered quarks on lattices with cutoff a=1/6T. Denoting the cross-over temperature by Tc,
we vary the temperature T from 0.89Tc to 1.92Tc, spanning both the hadronic and quark gluon
plasma phases. While chiral symmetry restoration in the vector channel appears to take place near
Tc, it is seen in the scalar channel only above 1.33 Tc. Varying spatial lattice sizes, we find very
little volume dependence in our results at 0.94 Tc. We discuss the stability of the scalar meson at
these temperatures.
The XXVIII International Symposium on Lattice Field Theory, Lattice2010
June 14-19, 2010
Villasimius, Italy
∗Speaker.
c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/
P
oS(Lattice 2010)168
Screening masses of mesons Debasish Banerjee
1. Introduction
When strongly interacting matter is heated, it is expected to go over to the Quark Gluon plasma
(QGP) phase. Lattice Quantum chromodynamics (QCD) is extensively used to study the properties
of matter across this transition. Even though the thermodynamic quantities and the equation of state
is known in sufficient detail, more information about the plasma phase is needed to distinguish it
from the hadronic phase.
The study of static correlation lengths constitute one such important clue. The low lying de-
grees of excitations of the plasma can be studied using the spatial correlation function of hadronic
operators [1]. The screening mass is obtained from the exponential decay of the screening corre-
lators at large spatial separations. Under certain analyticity assumptions, this screening mass can
be related to the pole of the real time propagator of the corresponding operator as argued in [1].
These correlation functions, and the screening masses encode information about the interactions in
the medium, indicating for example, whether certain symmetries broken at low temperatures are
restored at high temperatures. It is known that at very high temperatures the screening masses of
the mesons approach that of a free ideal gas, 2piT [2]. Moreover, finite volume corrections for the
thermodynamic quantities can also be studied using the screening masses.
We investigate the spatial correlation functions for hadrons in 2-flavour QCD with staggered
fermions in both the hadronic and the plasma phases. We study the pattern of chiral symmetry
restoration across the cross-over in the screening spectrum. We study the finite volume effects on
the correlation functions at low temperature and the possibility of the decay of the scalar meson.
2. Technical Details
The generation of the configurations used in the present study was reported in [3]. The lattices
had a temporal extent of Nt = 6 and spatial extent Ns = 24, except in the finite volume study where
spatial lattice sizes from 8 to 30 were used. Two flavours of light staggered quarks were used with
the bare quark masses tuned such that the T = 0, mpi ≈ 230 MeV at all temperatures T.
Point-to-point correlation functions for local meson operators in the pseudo-scalar (PS), scalar
(S), vector (V) and axial-vector (AV) channels were analyzed in detail. Denoting by G(x,0) the
staggered propagator, the correlation functions projected to zero momentum are:
C(z) = 1
N2x Nt
∑
x,y,t
Tr[G(x,0)G†(x,0)]gS(x) (2.1)
where gS(x) [4] are the staggered phase factors specific to the mesons in given quantum number
channels. We have calculated only the connected part of the correlation functions, which corre-
spond to the flavour non-singlet mesons. These correlation functions contain contributions from
the respective parity partner as well, and hence are parameterized as
C(z) = A1( e−m1z + e−m1(Nz−z) )+ (−1)zA2( e−m2z + e−m2(Nz−z) ) (2.2)
Here m1 and m2 are the screening masses of the lightest natural parity meson appropriate for the
operator used and its opposite parity partner. A1 and A2 are the respective normalizations. In this
convention, the Goldstone pion is the non-oscillating part of the correlator and has positive A1.
2
P
oS(Lattice 2010)168
Screening masses of mesons Debasish Banerjee
The tolerance of the conjugate gradient (CG) algorithm was chosen to be ε = 10−5. This en-
sured that the systematic error in the meson correlation functions arising due to the matrix inversion
was much lesser than the statistical errors.
The screening masses were extracted by fitting the correlation functions to the 2-mass fit form
in eqn 2.2. As usual, the correlation between the different z-slices are incorporated through the
covariance matrix and the fit values are obtained by minimizing the correlated-χ2 .
3. Results
While our main results are on the temperature dependence of the screening mass spectrum on
Nt = 6 lattices, and the consequent study of the chiral symmetry restoration pattern as a function of
temperature, we also present results for the spatial volume (in-)dependence of the screening masses
of all the mesons near the critical end-point temperature.
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
 0  2  4  6  8  10  12
|C(
z)|
z
PS
S
V
AV
10-6
10-5
10-4
10-3
10-2
10-1
100
101
 0  2  4  6  8  10  12
|C(
z)|
z
PS
S
V
AV
Figure 1: The correlators for all the mesons at T=1.92 Tc(left) and T=0.97 Tc(right). Note the prominent
oscillations for the V-AV mesons. We have plotted the absolute value of the correlators for the V and AV
mesons.
3.1 Screening spectrum
We display our results for all the meson correlators in fig 1 at two different temperatures: 1.92
Tc and 0.97 Tc. The V/AV correlators show prominent oscillations at both temperatures. For the
same statistics, there is more noise for the other mesons at large-z and lower temperature than the
pion, which is equally good at both temperatures. The pion correlation functions could be fit well
to a single mass form at all temperatures, except the highest temperature (1.92 Tc) where we had to
use the 2-mass fit. A full list of the screening masses will be given elsewhere[5].
To check the consistency of fitted masses, we also calculated the local masses. Due to the os-
cillating nature of the staggered fermions, the local masses were defined using correlators separated
by 2 z-slices as done in [4]:
C(z+ 1)
C(z−1) =
cosh[−m(z)(z+ 1−Nz/2)]
cosh[−m(z)(z−1−Nz/2)]
. (3.1)
3
P
oS(Lattice 2010)168
Screening masses of mesons Debasish Banerjee
The local masses so extracted are shown in Fig. 2 for the same two temperatures, 1.92 Tc and 0.97
Tc. The pion masses are the best determined in both the phases with small error bars and agree very
well with the fitted values. Such an agreement are also found for the other mesons, although with
larger bars.
 2
 4
 6
 8
 10
 12
 2  4  6  8  10
m
/T
z
PS
S
V
AV
 2
 4
 6
 8
 10
 12
 2  4  6  8  10  12
m
/T
z
PS
V
2 pi
Figure 2: The extracted local masses for the mesons at T=0.97 Tc(left) and T=1.92 Tc(right). For T=1.92 Tc,
only the PS and V meson masses are shown since the S and AV exactly coincide with them. The black line
coinciding with the lower blue line is the ideal gas limit.
The behaviour we discussed so far for both the hadronic and QGP phase is generic for other
temperatures as well, with gradual changes in the fit parameters. Fig. 3 shows the fits to the PS and
V correlators using the fitted values of the screening masses and corresponding amplitudes. The
correlators have been scaled to depict the entire range in temperature on the same plot. The 2-mass
nature of the fit used for the V correlators is very evident.
Finally, we plot the screening masses of the S, PS, V and AV mesons as a function of T/Tc
in Fig. 4. The chiral symmetry, broken at low temperatures, gets restored sufficiently away from
the cross-over temperature Tc. While the PS and S mesons masses become degenerate only around
1.33 Tc, the V and AV mesons are degenerate even at temperatures ≃ Tc. We note from Fig.2 that
while the V/AV screening masses appear to be the same as the free theory continuum limit at about
2 Tc, the screening masses of the PS/S mesons still differ from the ideal gas, and the V/AV mesons,
by about ∼ 20%. Our results confirm similar results obtained by other collaborations[2, 4, 6].
3.2 Finite Volume Study
A finite volume study was done at 0.94 Tc, since this is the phenomenologically interesting
temperature where the QCD CEP is expected to lie at µE = 1.6Tc [3]. A range of lattices with
Ns=8,12,18,24 and 30 were considered, where the smallest two had anisotropic spatial extents.
From fig. 5, it is clear that the screening masses do not have significant volume dependence. All
the lattices in our study are much bigger than the corresponding screening lengths of the mesons,
which explains the absence of the finite volume effect associated with stable states.
Further, given that the mass of the scalar meson is more than twice the mass of the pion, we
can discuss whether the decay of the scalar is possible. In the continuum, this decay is forbidden
4
P
oS(Lattice 2010)168
Screening masses of mesons Debasish Banerjee
10-4
10-3
10-2
10-1
100
101
102
103
104
105
 0  2  4  6  8  10  12
C(
z)
z
10-8
10-6
10-4
10-2
100
102
104
106
108
1010
 0  2  4  6  8  10  12
C(
z)
z
Figure 3: Single mass fits to the pseudo-scalar correlators (left) and the 2-mass fits to the vector correlators
(right) for the entire temperature range studied here. Correlators have been appropriately scaled to make
them fit in the same figure. The lines and symbols in each plot represent temperatures in units of Tc in
decreasing order from bottom: 1.92, 1.48, 1.33, 1.21, 1.01, 1.00, 0.99, 0.97, 0.94, 0.92, 0.89. (The lowest
curve is for 1.92 Tc and the topmost is for 0.89 Tc.)
 2
 3
 4
 5
 6
 7
 8
 9
 10
 11
 0.8  1  1.2  1.4  1.6  1.8  2
m
/T
T/Tc
Free continuum
PS meson
S meson
V meson
AVmeson
Figure 4: Masses of the PS, S, V and AV mesons a a function of T/Tc. The free continuum value of 2pi is
indicated with an arrow.
by parity. On the lattice however, unphysical decays are known to occur [7]. In our case, parity
and kinematical considerations imply that the scalar could only decay into the pion and one of
its taste partners. A finite volume study could potentially answer this interesting question. Our
results indicate that this decay does not occur. Fig 6 shows the scalar correlation functions are
quite featureless as a function of volume. Further support comes from the fitted and the measured
correlator normalization, CS(0), which are both independent of volume.
It is known that decays can be arrested if the volume is too small. In this context, we point out
that we have worked with reasonably large volumes. The LT values in our study range from 1.33
5
P
oS(Lattice 2010)168
Screening masses of mesons Debasish Banerjee
 2
 3
 4
 5
 6
 7
 8
 9
 10
 1  1.5  2  2.5  3  3.5  4  4.5  5  5.5
m
/T
LT
PS
S
V
AV
Figure 5: Masses of the PS, S, V and AV mesons a a function of LT at T = 0.94Tc.
10-6
10-5
10-4
10-3
10-2
10-1
100
101
 0  2  4  6  8  10  12  14  16
(-)
z  
C(
z)
z
Ns=8
Ns=12
Ns=18
Ns=24
Ns=30
 1
 1.5
 2
 2.5
 3
 3.5
 4
 1  1.5  2  2.5  3  3.5  4  4.5  5  5.5
C S
(0)
 in
 la
ttic
e u
nit
s
LT
Measured
Fitted coefficient
Figure 6: The upper figure shows the Scalar meson correlation functions. It is quite featureless as a function
of the volume. The lower figure shows that both the measured and fitted value of CS(0) is independent of
volume.
to 5 while mPSL range from 2.4 to 12.
4. Summary
We studied point-to-point correlation functions for mesons in the PS, S, V and the AV channels
for two flavour QCD in the staggered fermion formulation. We extracted the corresponding screen-
ing masses and showed that while the screening masses of the V/AV mesons become degenerate by
Tc, one needs to go up to 1.33 Tc to see the corresponding degeneracy of the PS/S mesons. More-
over, while by 2 Tc the V/AV screening masses are already seen to reach their continuum value,
the PS/S screening masses are still 15-20% away from the free theory value. Using a finite volume
study, we find that the scalar correlation functions are insensitive to the change in spatial volume,
6
P
oS(Lattice 2010)168
Screening masses of mesons Debasish Banerjee
which leads us to suspect that the scalar meson decay does not occur at finite lattice spacings at
temperatures less than the cross-over temperature.
Acknowledgments
The computations were performed on the CRAY X1 of the Indian Lattice Gauge Theory Ini-
tiative (ILGTI) in TIFR, Mumbai. D.B. wishes to acknowledge useful discussions with Saumen
Datta, Nilmani Mathur and Jyotirmoy Maiti.
References
[1] C. DeTar and J. B. Kogut Phys.Rev.D36(1987) 9
[2] K. Born et al (MTc Collaboration) Phys. Rev. Lett. 67, 302 (1991)
[3] R. V. Gavai, S. Gupta Phys.Rev.D78 (2008) 114503
[4] R. V. Gavai, S. Gupta and P. Majumdar Phys.Rev. D65 (2002) 054506
[5] D. Banerjee, R. V. Gavai and S. Gupta (in preparation)
[6] S. Mukherjee PoS LAT2007:210,2007
[7] C. Bernard et al Phys.Rev. D76 (2007) 094504
7


The scalar does not decay at finite temperatures

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The scalar does not decay at finite temperatures

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