Noncompact Lattice Simulations of SU(2) Gauge Theory

4 Pages·0.08 MB·English

by  Kevin Cahill

1
Noncompact Lattice Simulations of SU(2) Gauge Theory
Kevin Cahill
Department of Physics and Astronomy, University of New Mexico
Albuquerque, New Mexico 87131–1156,USA
3 Wilson loops have been measured at strong coupling, β = 0.5, on a 124 lattice in noncompact simulations of
9 pureSU(2) without gauge fixing. There is no sign of quark confinement.
9
1

n INTRODUCTION slow. One may develop faster code by interpo-
a
lating across plaquettes rather than throughout
J In1980Creutz[1]displayedquarkconfinement
simplices. For SU(2) such noncompact simula-
1 at moderate coupling in lattice simulations [2]
tions agree well with perturbation theory at very
2
of both abelian and nonabelian gauge theories.
weak coupling [9].
Whether nonabelian confinement is as much an
1 This report relates the results of measuring
v artifact of Wilson’s action as is abelian confine- Wilson loops at strong coupling, β ≡ 4/g2 =
0 ment remains unclear. 0.5, on a 124 lattice in a noncompact simulation
1 ThebasicvariablesofWilson’sformulationare
of SU(2) gauge theory without gauge fixing or
0
elementsofacompactgroupandentertheaction
1 fermions. CreutzratiosoflargeWilsonloopspro-
only through traces of their products. Wilson’s
0 vide a lattice estimate of the qq¯-force for heavy
3 action has extra minima [3]. Mack and Pietari- quarks. There is no sign of quark confinement.
9 nen[4]andGrady[5]haveshownthatthese false
t/ vacua affect the string tension. In their simula- THIS NONCOMPACT METHOD
a tions of SU(2), they placed gauge-invariant infi-
l
- nite potential barriers between the true vacuum In the present simulations, the action is free
p
and the false vacua. Mack and Pietarinen saw of spurious zero modes, and it is not necessary
e
h a sharp drop in the string tension; Grady found to fix the gauge. The fields are constant on the
: that it vanished. links of length a, the lattice spacing, but are in-
v
To avoid using an action that has confinement terpolatedlinearly throughoutthe plaquettes. In
i
X built in, some physicists have introduced lattice the plaquette with vertices n,n+eµ, n+eν, and
r actionsthatarenoncompactdiscretizationsofthe n+eµ+eν, the field is
a
continuum action with fields as the basic vari- x
ables [3, 6–11]. Patrascioiu, Seiler, Stamatescu, Aaµ(x) = aν −nν Aaµ(n+eν)
(cid:16) (cid:17)
Wolff, andZwanziger[6]performedthe firstnon- x
+ n +1− ν Aa(n), (1)
compactsimulationsofSU(2)byusingsimpledis- ν a µ
(cid:16) (cid:17)
cretizationsoftheclassicalaction. Theyfixedthe
and the field strength is
gauge and saw a force rather like Coulomb’s.
It is possible to use the exact classical action Fµaν(x) = ∂νAaµ(x)−∂µAaν(x)
in a noncompact simulation if one interpolates +gfaAb(x)Ac(x). (2)
bc µ ν
the gauge fields from their values on the ver-
ticesofsimplices[3,7,8]orhypercubes. ForU(1) TheactionS isthesumoverallplaquettesofthe
these noncompact formulations are accurate for integral over each plaquette of the squared field
general coupling strengths [8]. But in four di- strength,
mensionsandwithoutgaugefixing,suchformula- a2
tionsareimplementableonlyincodethatisquite S = 2 Z dxµdxνFµcν(x)2. (3)
Xpµν
2
The mean-value in the vacuum of a euclidean- MEASUREMENTS AND RESULTS
time-ordered operator W(A) is approximated by
a normalized multiple integral over the Aa(n)’s To measure Wilson loops and their Creutz ra-
µ tios, I used a 124 periodic lattice, a heat bath,
e−S(A)W(A) dAa(n) and 20 independent runs with cold starts. The
hTW(A)i ≈ µ,a,n µ (4) first run began with 25,000 thermalizing sweeps
0 R e−S(A) Q dAa(n)
µ,a,n µ at β = 2 followed by 5000 at β = 0.5; the other
R Q nineteen runs began at β =0.5 with 20,000 ther-
which one may compute numerically [1]. I used
malizing sweeps. In all I made 59,640 measure-
macsyma to write the fortran code [10].
ments, 20 sweeps apart. I used a version Parisi’s
trick [13] that respects the dependencies that oc-
CREUTZ RATIOS
cur in corners of loops and between lines sepa-
rated by a single lattice spacing. The values of
The quantity normally used to study confine-
the Creutz ratiosso obtainedare listedin the ta-
ment in quarkless gauge theories is the Wilson
ble along with the tree-level theoretical values as
loop W(r,t) which is the mean-value in the vac-
given by eqs.(7–8). I estimated the errors by the
uum of the path-and-time-ordered exponential
jackknifemethod[14],assumingthatallmeasure-
1 mentswereindependent. Binninginsmallgroups
W(r,t)= PT exp −ig AaT dx (5)
d (cid:28) (cid:18) I µ a µ(cid:19)(cid:29) made little difference.
0
divided by the dimension d of the matrices T
a Noncompact Creutz ratios at β =0.5
that representthe generatorsofthe gaugegroup.
r × t Monte Carlo Order 1/β
Although Wilson loops vanish [12] in the exact a a
theory, Creutz ratios χ(r,t) of Wilson loops de- 2×2 0.23107(4) 0.39648
fined [1] as double differences of logarithms of
3×3 0.03576(11) 0.13092
Wilson loops
4×4 0.00485(29) 0.06529
χ(r,t)=−logW(r,t)−logW(r−a,t−a)
5×5 0.00094(82) 0.03913
+logW(r−a,t)+logW(r,t−a) (6)
6×6 -0.00149(226) 0.02608
are finite. For larget,the Creutz ratioχ(r,t) ap-
proximates (a2 times) the force between a quark Ifthestaticforcebetweenheavyquarksisinde-
and an antiquark separated by the distance r. pendentofdistance,thentheCreutzratiosχ(r,t)
Fora compactLie groupwith N generatorsTa for large t should be independent of r and t.
normalized as Tr(T T ) = kδ , the lowest-order Themeasuredχ(r,t)’saresmallerthantheirtree-
a b ab
perturbative formula for the Creutz ratio is level perturbative counterparts. The measured
χ(r,t)’s also fall faster with increasing loop size.
N
χ(r,t) = [−f(r,t)−f(r−a,t−a) There is no sign of confinement.
2π2β
+f(r,t−a)+f(r−a,t)] (7) PLAUSIBLE INTERPRETATIONS
where the function f(r,t) is Why don’t noncompact simulations display
quark confinement? Here are some possible an-
r r t t
f(r,t) = arctan + arctan swers:
t t r r
a2 a2 1. Although noncompact methods have ap-
− log + (8)
(cid:18)r2 t2(cid:19) proximate forms of all continuum symme-
tries, including gauge invariance, they lack
and β is the inverse coupling β =d/(kg2). anexactlatticegaugeinvariance. Itmaybe
3
possible to impose a kind of lattice gauge Polyakov line of length 12a at β = 0.5 to
invariance by having the fields randomly be 0.00853(67) with the Haar measure and
make suitably weighted gauge transforma- 0.01135(1)without it.
tions of the compact form
7. Confinement is a robust and striking phe-
exp[−igaA′b(n)T ]= nomenon. Maybe the true continuum the-
µ b ory is one like Wilson’s that can directly
U(n+eµ)exp[−igaAbµ(n)Tb]U(n)† (9) account for it. The hybrid measure
or of some noncompact form. e− kDgc2f(c,l)Tr(1−ℜPeigHcAaµTadxµ)dµ(A)(11)
2. The Maxwell-Yang-Mills action contains R
squares of (covariant) curls of gauge fields, reduces to Wilson’s prescription if the
not squares of derivatives of gauge fields. weight functional f(c,l) of the path in-
Thus the gauge fields may vary markedly tegration over closed curves c is a delta
from one link to the next. In fact in these functional with support on the plaquettes
noncompactsimulations,the ratioofdiffer- and if dµ(A) incorporates the Haar mea-
ences of adjacentgaugefields to their mod- sure. A weight functional like f(c,l) ∼
uli exp[−(kck/l)4] where kck is the length of
the curve might give confinement for dis-
h|Abµ(n+eν)−Abµ(n)|i tancesmuchlongerthanl andperturbative
(10)
h|Ab(n)|i qcd for much shorter distances.
µ
exceedsunityforallcouplingsfromβ =0.5
ACKNOWLEDGEMENTS
to β = 60 and all µ and ν. It is not ob-
vious that noncompact methods can cope IamgratefultoH.Barnum,M.Creutz,G.Kil-
with such choppiness. cup, J. Polonyi,and D. Topa for useful conversa-
tions and to the Department of Energy for sup-
3. The noncompact lattice spacing a (β)
NC port under grant DE-FG04-84ER40166. Some of
is probably smaller than the compact one
the computations reported here were performed
a (β). Thus noncompactmethods may ac-
C incollaborationwithRichardMatzneroftheUni-
commodatetoosmallavolumeatweakcou-
versityofTexasatAustin. Someweredoneonan
pling; confinement might appear in non-
RS/6000 lent by ibm, some on Cray’s at the Na-
compact simulations done on much larger
tionalEnergyResearchSupercomputerCenterof
lattices or at stronger coupling. Both pos-
the Department of Energy, and some on a 710
sibilities would be expensive to test.
lent by Hewlett-Packard. Most were done on a
4. Perhaps SU(3), but not SU(2), confines. decstation 3100.
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