arXiv:hep-lat/9906023v2 2 Jul 1999
Screening Masses in SU(2) Pure Gauge Theory
The Mehta Research Institute,
Chhatnag Road, Jhusi, Allahabad 211019, India.
Department of Theoretical Physics,
Tata Institute of Fundamental Research,
Homi Bhabha Road, Mumbai 400005, India.
We perform a systematic scaling study of screening masses in pure
gauge SU(2) theory at temperatures above the phase transition tem-
perature. The major finite volume effect is seen to be spatial decon-
finement. We extract the screening masses in the infinite volume and
zero lattice spacing limit. We find that these physical results can be
deduced from runs on rather coarse lattices. Dimensional reduction is
clearly seen in the spectrum.
In a recent paper  we examined the spectrum of screening masses at
finite temperature in four dimensional SU(2) and SU(3) pure gauge theories.
Our primary result was that dimensional reduction could be seen in the
(gauge invariant) spectrum of the spatial transfer matrix of the theory. In
addition, we had shown that the specific details of the spectrum precluded
any attempt to understand it perturbatively. In this paper we present a
complete set of non-perturbative constraints on the effective dimensionally
reduced theory [2, 3, 4] at a temperature (T) above the phase transition
temperature (Tc) for the SU(2) case in the zero-lattice spacing and infinite
The study of screening masses is interesting for two reasons. First, they
are crucial to phenomenology because they determine whether the fireball
obtained in a relativistic heavy-ion collision is large enough for thermody-
namics. Second, the problem of understanding screening masses impinges
on several long-standing problems concerning the infrared behaviour of the
T > Tcphysics of non-Abelian gauge theories.
It is known that electric polarisations of gluons get a mass in perturbation
theory, whereas magnetic polarisations do not. Long ago, Linde pointed out
 that T > 0 perturbation theory breaks down at a finite order due to
this insufficient screening of the infrared in non-Abelian theories. The most
straightforward way to cure this infrared divergence would be if the magnetic
polarisations also get a mass non-perturbatively. There have been recent
attempts to measure such a mass in gauge-fixed lattice computations .
It was found long back that the solution could be more complicated
and intimately related to the dynamics of dimensionally reduced theories.
Jackiw and Templeton analysed perturbative expansions in massless and
super-renormalisable three dimensional theories  and found that subtle
non-perturbative effects screen the infrared singularities in such theories. In
a companion paper, Applequist and Pisarski discussed the possibility that
such effects might, among other things, also give rise to magnetic masses
. In fact the recent suggestion of Arnold and Yaffe that non-perturbative
terms and logarithms of the gauge coupling may be important in an expan-
sion of the Debye screening mass in powers of the coupling  may be seen
as an example of such non-perturbative effects. The generation of the other
screening masses are also non-perturbative. We discuss these issues further
after presenting our main results.
In this paper we report our measurements of the screening masses in the
infinite volume and zero lattice spacing limit of SU(2) pure gauge theory at
temperatures of 2–4Tc. We found a strong finite volume movement of one
of the screening masses due to spatial deconfinement. However, the lack of
finite volume effects in the remaining channels allowed us to extract infinite
volume results from small lattices. The effect of a finite lattice spacing turned
out to be small. We were able to pin down all the available screening masses
with an accuracy of about 5%.
It is necessary to set out our notation for the quantum numbers of the
screening masses. The transfer matrix in the spatial direction, z, has the
dihedral symmetry, D4
x, y and t directions. The irreducible representations (irreps) are labelled
by charge conjugation parity, C, the 3-dimensional (x,y,t) parity, P, and
the irrep labels of D4(four one-dimensional irreps A1,2, B1,2and one two-
dimensional irrep E). In SU(2) gauge theory, only the C = 1 irreps are
realised; hence we lighten the notation by dropping this quantum number.
Dimensional reduction implies the following pair-wise degeneracies of
hof a slice of the lattice which contains the orthogonal
1) = m(A−P
1) = m(B−P
2 ),m(EP) = m(E−P). (1)
After this reduction, the symmetry group becomes C4
in the continuum. The latter group has two real one-dimensional irreps— 0+
and 0−. The first comes from the Jz= 0 components of even spin irreps of
O(3), and the second from the Jz= 0 components of the odd spins. There
are also an infinite number of real two dimensional irreps, M, corresponding
to the Jz= ±M pair coming from any spin of O(3). Dimensional reduction
associates the irreps of D4
von the lattice and O(2)
hwith those of O(2) according to
m(0+) = m(A+
m(1) = m(E),
1),m(0−) = m(A−
m(2) = m(B+
1) = m(B−
The final double equality is valid only when all lattice artifacts disappear.
Although O(2) has an infinite tower of states, only these four masses are mea-
surable in a lattice simulation of the SU(2) theory3. Some of the equalities
in eqs. (1,2) may be broken by dynamical lattice artifacts.
3There has been a first attempt to disentangle these lattice effects and measure the
higher irreps .
We studied these artifacts using “torelon” correlators . These are
correlation functions of Polyakov loop operators in the spatial (Pxand Py)
and temporal (Pt) directions. Ptand Px+Pytransform as the scalar (A+
of finite volume effects in masses can be understood (for moderate mL) in
terms of torelons.
The status of the A+
at T = 0. Here, Pt is the order parameter for the phase transition, and
its correlations have genuine physical meaning— giving the static quark-
antiquark potential, and hence defining the Debye screening mass, MD. This
is identical to m(A+
respect, the finite temperature theory is nothing but a finite size effect.
We believe that the major part of finite volume effects in screening masses
can be understood in terms of finite temperature physics. In simulations of
Nt×L2×Nzlattices at a given coupling β, when the transverse direction, L,
is small enough, the spatial gauge fields are deconfined. The spatial torelons
Px,yare order parameters for this effect. In general, large lattices, L/Nt≫
T/Tchave to be used to obtain the thermodynamic limit. Below this limiting
value of L, we should find strong finite volume effects, but only in the A+
mass is expected to be twice the torelon mass. Whether or not similar effects
are seen in the A+
than MD/2. If it is, then finite volume effects should be strong, otherwise
not. We look upon torelons as convenient probes of finite volume effects, not
We have studied screening masses for SU(2) gauge theory on Nt×L2×Nz
lattices with Nz= 4Ntat a temperature of T = 2Tc. We studied two series of
lattices, one for Nt= 4 and another for Nt= 6. For the first, we took L = 8,
10, 12 and 16. For the second, we chose L = 16, 20 and 24. For Nt= 4,
a temperature of 2Tcis obtained by working with β = 2.51. On Nt = 6,
the choice β = 2.64 gives T = 2Tc. The choice of lattice sizes allowed us to
investigate finite volume as well as finite lattice spacing effects at constant
We have also carried out measurements at T = 3Tcand 4Tc. Since our
measurements at 2Tcshowed that lattice spacing effects are quite small for
Nt = 4, we restricted ourselves to this size at higher temperatures.
3Tc, we worked with a 4 × 243lattice. At 4Tc, we supplemented our earlier
h, whereas Px− Pytransforms as B+
1. At zero temperature, a major part
1torelon, Pt, at T > 0 is very different from that
1) obtained from the Wilson loop operators . In this
1sectors. When such effects can be directly measured, the B+
1sector depends on whether the spatial torelon mass is less
measurements on small (4 × 82× 16 and 4 × 122× 16) lattices  with
measurements on 4 × 243and 4 × 323lattices. For Nt= 4, temperatures of
3Tcand 4Tcare attained by working at β = 2.64 and 2.74, respectively.
We used a hybrid over-relaxation algorithm for the Monte-Carlo simula-
tion, with 5 steps of OR followed by 1 step of a heat-bath algorithm. The
autocorrelations of plaquettes and Polyakov loops were found to be less than
two such composite sweeps; hence measurements were taken every fifth such
sweep. We took 104measurements in each simulation except on the 6 × 243
lattice where we took twice as much, and the 4 × 323lattice where we took
Noise reduction involved fuzzing. The full set of loop operators measured
on some of the smaller lattices can be found in . Since analyses of subsets
of these operators gave identical results, we saved CPU time on the larger
lattices by measuring a smaller number of operators. The full matrix of cross
correlations was constructed, between all operators at all levels of fuzzing, in
each irrep. A variational procedure was used along with jack-knife estimators
for the local masses. Torelons were also subjected to a similar analysis.
Our measurements at 2Tcfor Nt= 4 are reported in Table 1. We can
measure torelons for fairly large values of L/Nt. Twice the A+
screening mass is greater than that obtained from Pt. Hence m(A+
from loops is equal to the latter and therefore shows no finite volume effect.
screening mass closely equals twice the B+
a systematic dependence on L. Finite volume effects are absent in all the
other channels, as expected. For the L = 16 lattice for Nt= 4, the torelon is
not measurable, and finite volume effects are under control. At this largest
volume dimensional reduction and continuum physics are visible since the
equalities in eqs. (1,2) are satisfied.
We have investigated finite lattice spacing effects by making the same
measurements at the same physical temperature on lattices with Nt= 6. The
measurement of m(E−) turns out to be rather noisy. Since we had observed
on the coarser lattice that m(E+) = m(E−), we saved on CPU time by
dropping the measurement of the E−screening mass on the Nt= 6 lattices.
Our results on the finer lattice are collected in Table 2. Again, dimensional
reduction and continuum physics is visible because the equalities in eqs. (1,2)
are satisfied on the largest lattice.
From the data collected in Tables 1 and 2 it is clear that the physical ratio
1spatial torelons have equal screening masses. The B+
1 torelon mass, and hence shows
L = 8
0.71 ± 0.05
1.14 ± 0.02
1.18 ± 0.03
1.9 ± 0.2
1.8 ± 0.1
1.9 ± 0.2
L = 10
0.73 ± 0.06
1.01 ± 0.03
1.45 ± 0.05
1.9 ± 0.1
1.8 ± 0.1
1.9 ± 0.2
L = 12
0.69 ± 0.04
1.02 ± 0.02
1.62 ± 0.06
1.8 ± 0.1
1.75 ± 0.05
1.8 ± 0.2
1.80 ± 0.07
1.68 ± 0.07
0.80 ± 0.02
0.9 ± 0.1
0.8 ± 0.1
L = 16
0.73 ± 0.05
0.99 ± 0.02
1.73 ± 0.08
1.8 ± 0.1
1.76 ± 0.09
1.9 ± 0.2
1.8 ± 0.1
1.7 ± 0.2
0.73 ± 0.08
Torelon0.76 ± 0.04
0.46 ± 0.02
0.48 ± 0.02
0.75 ± 0.05
0.7 ± 0.1
0.7 ± 0.1
Table 1: Values of m on 4 × L2× 16 lattices in units of inverse lattice
spacing, 1/a = 4T at T = 2Tc. Local masses could be followed to distance
z ≈ 4/m. Blanks in the table mean that the operators were not measured,
and an entry of “-” denotes that the measurements were too noisy to yield a
L = 16
0.46 ± 0.02
0.69 ± 0.01
0.98 ± 0.03
1.23 ± 0.06
1.23 ± 0.08
1.28 ± 0.09
1.25 ± 0.03
0.46 ± 0.02
0.32 ± 0.17
0.37 ± 0.16
L = 20
0.51 ± 0.01
0.68 ± 0.02
1.07 ± 0.06
1.17 ± 0.07
1.23 ± 0.09
1.27 ± 0.05
1.3 ± 0.1
0.52 ± 0.02
L = 24
0.51 ± 0.02
0.68 ± 0.02
1.08 ± 0.09
1.19 ± 0.08
1.22 ± 0.06
1.25 ± 0.11
1.24 ± 0.08
0.51 ± 0.02
Table 2: Values of m on 6 × L2× 24 lattices in units of the inverse lattice
spacing 1/a = 6T at T = 2Tc. Local masses could be followed to distance
z ≈ 4/m. An entry of “-” in the table denotes that the measurements were
too noisy to yield a screening mass.
m/T is the same with both lattice spacings, for loop masses. Hence finite
lattice spacing effects are under control. This result is consistent with zero
temperature lattice measurements which show that at these lattice spacings,
ratios of physical quantities are independent of the spacing.
In Figure 1 we illustrate the nature of the finite volume effects. The lack
of movement in m(A+
fact of dimensional reduction to prune the amount of data that has to be
displayed in the graph. Note that the data show that m(2) can be estimated
by measuring any of the B irrep screening masses, apart from the B+
small volumes. Note also that m(B+
However, for Nt= 6, it becomes difficult to measure the torelon correlator
at fairly small value of L/Nt. This supports our earlier statement that the
torelon is a measure, not the cause, of finite volume effects.
In the SU(3) pure gauge theory, which has a first order phase transi-
tion, the simple equalities Nz/Nt,L/Nt> T/Tcare sufficient to remove finite
volume effects . The observed slow finite volume movement of m(B+
special to the SU(2) gauge theory, which has a second order finite tempera-
ture phase transition. As a result, the above constraints on the lattice sizes
1) and m(B+
2) is obvious. We have used the
1)/T scales with either L/Ntor L/Nz.
Figure 1: Values of m/T using different operators as a function of L/Nt(or
L/Nz), the scaled transverse spatial size of the lattice. The open symbols are
for Nt= 6 lattices, filled symbols for Nt= 4. In order to improve visibility,
the data at L/Nt= 4 for Nt= 6 has been displaced slightly to the right.
are compounded by two separate systematics of second order phase tran-
sitions. The first is that there are precursor effects which cause masses to
decrease at temperatures less than Tc; the second that part of this decrease
is power-law singular in Nzat fixed β. Consequently, in the SU(2) theory
we can at best state the more stringent conditions L/Nt= Nz/Nt≫ T/Tc.
In our measurements with Nt× N3
found that the lattice artifact in m(B+
Ns/Ntthat we had. At both the higher temperatures no other finite volume
effects were seen within the precision of the measurements. As a result, we
were able to estimate all the four screening masses listed in eq. (2). The
ratios m/T are seen to be approximately constant in this temperature range.
This is illustrated in Fig. 2.
The four masses that we have extracted from simulations of the 4-d theory
represent the maximum information available non-perturbatively to constrain
the effective 3-d theory. We found it instructive to display the same data
slattices at higher temperatures, we
1) persists for the largest values of
Figure 2: Values of m/T for 2Tc ≤ T ≤ 4Tc, showing the near-constancy
of the ratios in this temperature range. In order to improve visibility, the
points for m(1)/T have been shifted slightly to the left and those for m(2)/T
slightly to the right.
in Figure 3 as a plot of the ratio m(0+)/m(2) against m(0−)/m(2). The
finite volume movement in these numbers is fairly large if the denominators
are estimated through m(B+
reduced if m(B−
thermodynamic limit is pretty well pinned down, the figure also serves well
to compare the 4-d theory with different 3-d theories.
The point for the 3-d SU(2) pure gauge theory in the infinite volume and
for zero lattice spacing  is shown in the figure. It is clear that this is not
the appropriate effective theory. This result is expected, since a perturbative
mode counting shows that the effective three dimensional theory must con-
tain a gauge field and a scalar field that transforms adjointly under gauge
transformations, and the scalar field does not decouple completely from the
theory even at high temperatures .
The vertical bands in Figure 3 come from measurements in a 3-d SU(2)
gauge theory with a fundamental scalar in both the symmetric and Higgs
1). However, as shown, the movement is much
1) is used as an estimator of m(2). Since the continuum and
Figure 3: Finite size movement of ratios of screening masses with two different
identifications of the screening mass m(2). The physical point shown should
be understood to have error bars compatible with those for the other points.
The point for the 3-d pure gauge theory  is denoted by the cross. The
vertical bands show the measured 1−σ ranges for the mass ratio m(0+)/m(2)
in an SU(2) scalar-gauge theory  in the symmetric and Higgs phases.
phases of this theory . It is not surprising that the ratio m(0+)/m(2) in
either phase of this theory does not agree with our measurements at 2Tc, in
view of the arguments already presented.
In  a super-renormalisable 3-d theory of SU(2) gauge fields and an
adjoint scalar, with three couplings, was suggested as the effective theory.
Matching two of these couplings in a perturbation expansion, the screening
masses were computed through a simulation of the 3-d theory. It turned
out that at couplings corresponding to a temperature of 2Tc, m(1)/m(0+) =
1.6±0.2 as opposed to the value 2.4±0.2 that we measure. Whether better
agreement can be obtained by fine-tuning the third coupling remains as a
future exercise. If three couplings can be tuned to reproduce four masses,
then this would vindicate the perturbative approach to matching espoused
However, until such a demonstration is made, there are questions whether
this procedure is viable at T ≈ 2Tc. A direct measurement suggests that the
gauge coupling is larger than unity, g2/2π ≈ 0.53, even for T = 2Tc. A
related statement has been made based on a recent study of the Debye mass—
that higher orders in the perturbative series become numrically smaller only
at T ≈ 107Tc . A similar statement comes from attempts to find the
region of validity of the perturbative expansion of the free energy in a non-
Abelian plasma , which give T > 105Tc. It has recently been suggested
 that effects associated with screening and damping should be resummed
to all orders in g, if perturbation theory is to behave reasonably at T ≈ 2Tc.
We have earlier concluded that the screening masses we observe cannot be
obtained perturbatively . The fact that our measurements show m/T > 2π
in some channels also indicates that the perturbative matching procedure
may not be useful, since dimensional reduction works only if modes with
energy 2πT or more decouple [2, 3].
There are alternatives to perturbation theory. One interesting method
would be to use gauge invariant composites directly to construct the effective
theory. Phenomenology of this kind was used long ago to examine lattice
data on the energy density for T > Tc SU(3) gauge theory . A more
sophisticated attempt of this kind was tried in , but needed the machinery
of large-N theories to control the expansion.
The question of the compositeness of screening masses is closely related.
Note that the 3-d adjoint Higgs, At, is in the 0−irrep of O(2), and the 3-d
gauge field, A, in the 1. The gauge invariant 0+can be seen in correlations
of the composite operators O1 = Tr(A2
higher dimensional operators. The gauge invariant 0−screening mass can be
seen, for example, in correlations of O3= Tr(A3
corresponding to the remaining gauge invariant screening masses can also be
easily written down. Nadkarni had shown by explicit computation that O1
and O2mix at order g4, where g is the 4-d gauge coupling . Hence, the
characterisation of m(0+) as being due to electric phenomena is a perturba-
tive statement, and is more or less correct according to how large g is at the
temperature that concerns us. Similar problems occur in the other channels
In summary, we identified the only source of large finite volume effects in
the determination of screening masses at T > Tcin SU(2) pure gauge theory.
These are due to spatial deconfinement and can be conveniently studied using
t) and O2 = Tr(A · A), as well as
t). The composite operators
torelons. Finite lattice spacing effects turn out to be easy to control. We
found that rather small and coarse lattices can be used to obtain a good
measurement of the physical screening masses, provided one ignores the B+
channel. Our best estimates are shown in Figure 2. Dimensional reduction,
as expressed non-perturbatively in eqs. (1,2), is seen in the temperature range
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