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arXiv:hep-lat/9906023v2 2 Jul 1999

MRI-PHY/P9906017

TIFR/TH/99-30

June, 1999

hep-lat/9906023

Screening Masses in SU(2) Pure Gauge Theory

Saumen Datta1

The Mehta Research Institute,

Chhatnag Road, Jhusi, Allahabad 211019, India.

and

Sourendu Gupta2

Department of Theoretical Physics,

Tata Institute of Fundamental Research,

Homi Bhabha Road, Mumbai 400005, India.

Abstract

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.

1E-mail: saumen@mri.ernet.in

2E-mail: sgupta@theory.tifr.res.in

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In a recent paper [1] 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

volume limit.

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

[5] 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 [6].

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 [7] 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

[8]. 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 [9] 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

1

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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

screening masses—

hof a slice of the lattice which contains the orthogonal

m(AP

1) = m(A−P

2 ),m(BP

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),

1) = m(B−

1).(2)

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 [10].

2

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We studied these artifacts using “torelon” correlators [11]. These are

correlation functions of Polyakov loop operators in the spatial (Pxand Py)

and temporal (Pt) directions. Ptand Px+Pytransform as the scalar (A+

D4

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+

and B+

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

their cause.

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

physics.

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

1) of

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 [12]. In this

1

1sectors. When such effects can be directly measured, the B+

1loop

1sector depends on whether the spatial torelon mass is less

At

3

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measurements on small (4 × 82× 16 and 4 × 122× 16) lattices [1] 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

4000.

Noise reduction involved fuzzing. The full set of loop operators measured

on some of the smaller lattices can be found in [1]. 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.

The A+

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 torelon

1) obtained

1and B+

1spatial torelons have equal screening masses. The B+

1 torelon mass, and hence shows

1loop

4