Dimensional reduction and screening masses in pure gauge theories at finite temperature
ABSTRACT We studied screening masses in the equilibrium thermodynamics of SU(2) and SU(3) pure gauge theories on the lattice. At a temperature of 2Tc we found strong evidence for dimensional reduction in the non-perturbative spectrum of screening masses. Mass ratios in the high-temperature SU(3) theory are consistent with those in the pure gauge theory in three dimensions. At the first-order SU(3) phase transition we report the first measurement of the true scalar screening mass.
arXiv:hep-lat/9806034v2 13 Sep 1998
Dimensional Reduction and Screening Masses
in Pure Gauge Theories at Finite Temperature
Saumen Datta1and Sourendu Gupta2
Department of Theoretical Physics,
Tata Institute of Fundamental Research,
Homi Bhabha Road, Mumbai 400005, India.
We studied screening masses in the equilibrium thermodynamics of
SU(2) and SU(3) pure gauge theories on the lattice. At a temperature
of 2Tcwe found strong evidence for dimensional reduction in the non-
perturbative spectrum of screening masses. Mass ratios in the high
temperature SU(3) theory are consistent with those in the pure gauge
theory in three dimensions. At the first order SU(3) phase transition
we report the first measurement of the true scalar screening mass.
The equilibrium thermodynamics of a gauge theory is studied in the non-
perturbative domain by lattice simulations of the partition function. Much
is now known about the phase transitions in SU(2) and SU(3) pure gauge
theory and in QCD, including the order, the transition temperature, Tc,
entropy density, pressure, specific and latent heats and other such quantities
Also of interest are the screening masses at finite temperatures. These
are defined in general by the exponential spatial falloff of the correlation of
two static sources. A classification of all masses is provided by the transfor-
mation properties of the sources. For glueball-like screening masses, such a
classification was performed in  where the first measurements of several of
these masses were reported. Extensive lattice data is available on screening
masses from meson and baryon-like sources , and on the correlation of
A gauge invariant transfer matrix formulation is easy to write for the
lattice regularised theory. For thermal physics, it is convenient to think of
the transfer matrix in a spatial direction. The free energy and other bulk
thermodynamic quantities involve only the largest eigenvalue of the trans-
fer matrix. Correlation functions and the screening masses involve the ratio
of this largest eigenvalue with specific other eigenvalues depending on the
transformation properties of the source. Thus, the full spectrum of screening
masses contains much more information about the theory than bulk thermo-
dynamics can provide.
A quantity of special interest is the Debye screening mass, MD, whose
inverse gives the screening length of static choromo-electric fields. In the
quantum theory, this screening mass plays an important role in regulating
some infra-red singularities. Lattice measurements have, in the past, con-
centrated on measuring a mass from the correlation of Polyakov loops, MP.
It was expected that when the gauge coupling becomes small, MP = 2MD.
Many years ago Nadkarni showed that the relation between MDand MP is
far from being so simple . Subsequent lattice work focussed on methods
of computing MD.
Reisz and collaborators [5, 6] wrote down a dimensionally reduced theory
which could be used to define MD non-perturbatively. A recent paper 
used the representations of the symmetries of the transfer matrix to write
down operators whose correlations could be used to measure the Debye mass3
and gave a general parametrisation of the perturbative and non-perturbative
terms for MD. These parameters have since been determined in a lattice
measurement of the Debye screening mass using a dimensionally reduced
theory at very high temperatures .
A similar screening mass is required for the magnetic sector of the non-
abelian gauge theory in the plasma, in order to get sensible results in the
infrared. However, the magnetic mass is entirely non-perturbative in nature.
It has been the object of many lattice studies [9, 10]. In fact, a recent work
 tries to extract MDand Mmfrom gauge fixed gluon propagators at finite
Many years ago Linde discovered  that the thermal perturbation ex-
pansion in non-Abelian gauge theories breaks down because the magnetic
sector is not amenable to perturbative studies. A recent attempt to under-
stand Linde’s problem in the region where the coupling g ≪ 1 has invoked a
sequence of dimensionally reduced effective theories . At length scales of
1/gT dimensional reduction yields a three dimensional SU(N) gauge theory
coupled to an adjoint scalar field of mass MD ≈ O(gT). At longer scales,
1/g2T, the scalar field can be integrated out, and the leading terms in the
effective theory correspond to a pure gauge theory in three dimensions. On
the basis of this reduction it has been argued [12, 7] that a non-vanishing
pole in magnetic gluon propagators is absent, and Linde’s problem is finessed
by confinement in the three dimensional gauge theory.
In fact, dimensional reduction has often been used to explore finite tem-
perature theories , and it has long been argued that the dimensionally
reduced theory is confining. The spatial string tension is known to be non-
vanishing for T > Tc and scales as T2. This, and other, lattice mea-
surements reveal that the gauge coupling close to Tcare too large to trust
At temperatures of a few Tc, the coupling g ≥ 1, and the length scales
1/T ≈ 1/gT ≈ 1/gT2, and hence cannot be decoupled. The construction
of  is not applicable. However, the transfer matrix argument assures us
that MDis only the smallest in a hierarchy of screening masses. We can then
use the spectrum of the screening masses to explore the symmetries of the
transfer matrix and check whether or not dimensional reduction works; and
3The group theoretical identification of MP and MDwas, in fact, first given in .
if it does, then what is the nature of the reduced theory.
Our major result is that the degeneracies of the spectrum of screening
masses implies that the symmetry group of the spatial transfer matrix is that
of two dimensional rotations— implying dimensional reduction for T ≈ 2Tc.
The measured non-perturbative spectrum is similar to that in 3-d pure gauge
The symmetries of the transfer matrix, and the physical consequences
are discussed in Section 2. The group theory presented in this section is
central to the rest of this paper. The extraction of screening masses in T > 0
four-dimensional SU(3) and SU(2) pure gauge theories take up the next two
sections. These may be skipped by those readers who are not interested in
the details of the lattice simulations. Section 5 contains a summary of our
lattice results and presents our conclusions on the nature of the dimensionally
reduced theory. Several technical details are relegated to appendices. Ap-
pendix A gives the loop operators used in our computations. In Appendix
B, we examine the possibility of explaining the mass spectrum in terms of
perturbative multigluon states. Noise reduction techniques for the lattice
simulations and the algorithm for projecting on to the lowest state in every
channel are described in Appendices C and D respectively.
2 Symmetries of the Transfer Matrix
For the T = 0 continuum Euclidean theory the symmetry of the transfer
matrix is the direct product of the full rotation group O(3) and the Z2groups
generated by charge conjugation, C, and time reversal. Irreps of O(3) are
labelled by the angular momentum and parity, JP, and of the full symmetry
group by JPC. For the lattice regularised theory, O(3) breaks to the discrete
subgroup of the symmetries of the cube, Oh. The consequent reduction of
the irreps of O(3) is well-known .
In the Euclidean formulation of the (continuum) equilibrium T > 0 the-
ory, the transfer matrix in one of the spatial directions is invariant under
symmetries of the orthogonal slice. Such slices are three dimensional— two of
which are spatial and one is the Euclidean time. The symmetry group is that
of a cylinder, C = O(2) × Z2. This Z2factor is generated by σz : t ?→ −t.
The non-Abelian group O(2) contains the Abelian subgroup of rotations,
SO(2), and a 2-d parity, Π : (x,y) ?→ (x,−y). The 3-d parity P = C(π)σz,
where C(π) is the rotation by π in SO(2).
In the high temperature limit, the effective theory is expected to undergo
dimensional reduction to a 3-d gauge theory. In such a theory, the transfer
matrix must have the O(2) symmetry of a 2-d slice. O(2) has two one-
dimensional irreps 0±(the 2-d scalar and pseudo-scalar) and an infinite tower
of two-dimensional irreps M. Under the reduction of SO(3) to O(2), the spin
J irrep breaks as—
J → 0Π(J)+
M,where Π(J) = (−1)J.(2.1)
On the lattice the irreps of C break further into irreps of the automor-
phism group of a z-slice, the tetragonal group D4
three dimensional lattice theory, the automorphism group of the 2-d slice is
classification of the states .
This pattern of symmetry breaking is summarised by
h= D4× Z2(P). For the
v, which is isomorphic to D4. A recent work on 3-d glueballs used this
O(3) = SO(3) × Z2(P)
−→ Oh= O × Z2(P)
C = O(2) × Z2(σz)
h= D4× Z2(P)
Four of the Z2factor groups are identical, and generated by the 3-d parity
P. O(2) and C4
vcontain the 2-d parity Π.
2.2 Point group representations
In this paper we use the notation of  for the crystallographic point groups
and their irreducible representations (irreps). This subsection contains a
discussion of the irreducible representations of the lattice symmetries.
The group D4is generated by eight elements in five conjugacy classes—
the identity (E), rotations of ±π/2 around the z-axis (C4), rotations of π
around the z-axis (C2
4), around the x or y-axes (C2) and around the directions