Table Of ContentConfinement, Topology,
and Other Non-Perturbative Aspects of QCD
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Series II: Mathematics, Physics and Chemistry - Vol. 83
Confinement, Topology and
Other Non-Perturbative Aspects of
QCD
edited by
Jeff Greensite
San Francisco State University,
San Francisco, CA, U.S.A.
and
Stefan Olejnik
Institute of Physics,
Slovak Academy of Sciences,
Bratislava, Slovakia
Springer-Science+Business Media, B.V.
Proceedings of the NATO Advanced Research Workshop on
Confinement, Topology, and Other Non-Perturbative Aspects of QCD
Starä Lesnä, Slovakia
21-27 January, 2002
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4020-0874-0 ISBN 978-94-010-0502-9 (eBook)
DOI 10.1007/978-94-010-0502-9
Printed on acid-free paper
All Rights Reserved
©2002 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2002
Softcover reprint of the hardcover 1st edition 2002
No part of this work may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming,
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Contents
Preface vii
P. van Baal/ Chiral zero-mode for abelian BPS dipoles 1
M. Baker, R. Steinke / Dual superconductivity, effective string theory,
and Regge trajectories 11
V. A. Belavin, M. N. Chemodub, M. I. Polikarpov / Confinement-
deconfinement order parameters 23
O. Borisenko, S. Voloshin, M. Faber / Analytical study of low
temperature phase of3DLGT in the plaquette formulation 33
F. Brockmann / Monopoles from instantons 41
M. Buballa, M. Oertel/Colorsuperconductivity in two- and three-flavor
systems at moderate densities 49
M. N. Chemodub, K. Ishiguro, T. Suzuki / Lattice monopoles in hot
SU(2) gluodynamics as blocked continuum defects 61
A. Di Giacomo / Color confinement and dual superconductivity: An
update 69
L. Dittmann, T. Heinzl, A. Wipf/Aneffectivetheory for theSU(2)mass
gap? 83
B. P. Dolan, D. O'Connor, P. Presnajder / Matrix models on the fuzzy
sphere 91
D. Dudal, K. Van Acoleyen, H. Verschelde /Dynamical massgeneration
in quantum field theory: Some methods with application to the
Gross-Neveu model and Yang-Mills theory 97
M. Engelhardt / Center vortex model for nonperturbative strong
interaction physics 105
M. Faber,J. Greensite, S. Olejnik/Status ofcenterdominance in various
centergauges 117
M. Faber, J. Wabnig, A. N. Ivanov /Solitons and spontaneous symmetry
breaking in 2 and 4 dimensions 129
Ph. de Forcrand, O. Jahn /50(3) versus 5U(2) lattice gauge theory 137
vi
F. Freire / 5U(2) Abelian projected action for renonnalisation group
flows 149
F. Gliozzi, M. Panero, P. Provero / Center vortices, magnetic condensate
andconfinement in a simple gauge system 157
M. Goltennan, Y. Shamir/ Lattice chiral gauge theories through gauge
fixing 165
A. Gonzalez-Arroyo / Spectra of lattice Dirac operators in non-trivial
topology backgrounds 177
1. Greensite / The gluon chain model revisited 185
R. W. Haymaker, A. Alexandru / Connections between thin, thick and
projection vortices in SU(2) lattice gauge theory 197
R. Horan, M. Lavelle, D. McMullan, A. Khvedelidze / Charges,
monopoles and gauge invariance 205
1. Horvath, S. J. Dong, T. Draper, F. X. Lee, H. B. Thacker, 1. B. Zhang
/ Low-lying Dirac eigenmodes, topological charge fluctuations and
the Instanton Liquid Model 213
J. Hosek / Anisotropic QCD superfluids 225
T. G. Kovacs / Towards the topological susceptibility with overlap
fennions 233
T. G. Kovacs, E. T. Tomboulis / Vortex waistlines 241
K. Langfeld / Vortex induced confinement and the Kugo-Ojima
confinementcriterion 253
H. Markum, R. Pullirsch, W. Sakuler / Monopoles and Lyapunov
exponents in U(l) lattice gauge theory 261
~. Martinovic / Massive lightfront QED(l+l) in the Weyl gauge 269
H. Reinhardt/ Topology ofcenter vortices 277
L. von Smekal, Ph. de Forcrand/ Electric flux sectors and confinement 287
1. D. Stack,W. W. Tucker, R. 1. Wensley /Confinementin SU(3): Simple
and generalizedmaximal abelian gauge 295
1. Wosiek / Supersymmetric Yang-Mills quantum mechanics 303
V. 1. Zakharov / Scalar-particle projection ofgluodynamics: Fine tuning 315
ListofParticipants 323
Preface
The NATO Advanced Research Workshop on Confinement, Topology, and
OtherNon-Perturbative Aspects ofQCD was held in January 2002 in the village
of Stani Lesmi, which lies in the mountainous High Tatra region of the Slovak
Republic. The meeting brought together forty-five participants from Europe and
the United States to discuss issues and present new results in an area of strong
interaction physics known as "confinementand topology."
Thequarkconfinementproblemremainsoneoftheclassic unsolved problems
ofhigh-energy physics, datingfrom theformulation ofquantum chromodynamics
as the theory ofthe strong nuclearforce in theearly 1970's. Few people seriously
doubtthatquarksareconfinedinhadrons; theabsenceoffreequarks inaccelerator
and Millikan-type experiments is very persuasive on this point. Noristhere much
doubt that the heavy quark potential increases linearly at large distances, if the
string-breaking effects of light quarks are discounted. Monte Carlo simulations
oflattice QCD have provided us with convincing evidence ofthe linear behavior
of the confining potential. But to this day there is no general agreement on the
mechanism which accounts for the confining force. Many physicists believe that
theQCD vacuum isdominated by some special class offield configurations, with
particular topological properties, whose influence at largedistance scales induces
quark confinement. However, there is no consensus on exactly which topological
objects dominate the vacuum at large scales. Monopoles and center vortices are
the most popular candidates for this role (and were well represented in Stara
Lesna), but large instantons and merons have also had their advocates. In recent
years there has been much progress towards subjecting proposed confinement
mechanisms, based on various topological objects, to numerical tests.
Our intention in organizing this Advanced Research Workshop was to bring
togethermanyoftheexperts inthisfieldtopresenttheirrecentworkandtodiscuss
witheach otherthe latestdevelopments. Apartfrom theconfinementproblem, we
also heard contributions on related non-perturbative issues such as chiral gauge
theories, color superconductivity, and the relevance (or perhaps irrelevance) of
instantons to chiral symmetry breaking. We were gratified that there were many
lively discussions, bothduring and long aftertheofficial seminarhours, on issues
that were raised in the talks.
We would like to thank the members of the local organizing committee for
theirassistance inpreparingandrunning themeeting. WearegratefultotheNATO
Science Programme for financial support. Most ofall we would like to thank the
participants, for their stimulating scientific contributions which are collected in
this volume.
JeffGreensite andStefan OlejnzK
vii
CHIRAL ZERO-MODEFOR ABELIANBPS DIPOLES
PIERREVAN BAAL
Instituut-Lorentzfor Theoretical Physics
University ofLeiden, P.D.Box9506
NL-2300 RA Leiden, The Netherlands
Abstract. We present an exact normalisable zero-energy chiral fermion solution for abelian BPS
dipoles. Forasingledipole,thissolutioniscontainedwithinthehightemperaturelimitoftheSU(2)
caloronwithnon-trivial holonomy.
1. TheDirac Monopole
Aconvenient representation for the Dirac monopole [1] is given by
A=gnAVlogw(x), w(x)=lxl+x·n, (1)
-n,
where w(x) is positive, but vanishes along the Dirac string pointing along as
seen from the monopole. The magnetic field is
B= VAA = -gV (n. VIOgw(x)) +gn~logw(x) = Breg+Bstn (2)
and using thatn·Vlogw(x) = 1/Ixl (thederivativealong thedirection oftheDirac
string) is independent of n, the first term gives rise to the radial magnetic field
associated with a magnetic point charge, whereas the second term represents the
Dirac string, which takes care of the return flux. This follows from the fact that
logw(x) is harmonic, except where w(x) vanishes. To be specific, choosing for
convenience n = e3 = (0,0,1), we find B= gx/lxl3+4nge38(x)8(y)8(-z) and
V.B V.B
reg = - str = 4ng83(x). Itgives the appropriate magnetic pointcharge for
B V.B
reg,but when including the return flux = 0, as it should.
Thefunction logw(x) canbeviewedas apotential, although from the pointof
view ofthe Maxwellequations itis morenatural toconsiderAo== - gn·V'logw(x)
as such. Not only B = VAo, but also as the time component of the Euclidean
reg
vector potential this choice of Ao gives rise to a self-dual configuration, with
E= VAo = Breg. The usefulness of w(x) becomes clear when one considers the
1
1. GreensiteandS.Olejnfk(eds.),
Confinement. Topology, andOtherNon-PerturbativeAspectso/QCD, 1-9.
©2002KluwerAcademicPublishers.
2
massless (Euclidean) Dirac equation in such a background. As usual we split this
into positive and negative chirality Weyl equations,
H+ --D---Dt -- -u~·DJ1' (3)
where DJ1 =dJ1+ieAJ1 is the covariant derivative and aj =i'tj' whereas ao is the
2 x 2 identity matrix. Forft = e3 a solution ofD'¥ = 0 is given by
(4)
Ifsodesired a (spin-)rotation allows one to obtain the solution for arbitrary ft, but
e3,
to keep things simple we stickto ft = such that
+
-dl :.'I i(2) Iogw(-) (5)
X •
-°3
Using the Dirac quantisation condition eg = 1,oneeasily verifies
-~IO~W(X))
(6)
-iH+'¥(x) = Jw(x) ( .
Since JW vanishes along the Dirac string, we find that JW~logw= 0 (as a dis
tribution), hence D'¥ = O. Likewise 1'¥12 = wVlogw. Vlogw = ~w- w~logw.
Thus 1'¥12 =~w =2/lxl has an integrable singularity at the origin, and the Dirac
string is invisible, as it should. Nevertheless, '¥ does notdecay sufficiently fast to
be normalisable. Note that the zero-mode is time independent. Putting do to zero,
H is precisely the Dirac Hamiltonian, with Ao playing the role of a Higgs field.
A non-zero asymptotic value ofAo would lead to a mass scale and exponentially
decaying wavefunctionsI.
It should not come as a surprise that existence of zero-energy solutions is
sensitive to the sign ofthe electron charge (relative tog). With eg = -! we have
(~ ~ (~IOgow(X)).
-iH+ (:.'I d3 IOgW(X)) = (7)
w(x) VI +1.:.0'12) w(x)
However, here the singularity of the Dirac string is no longer nullified but en
hanced. Nevertheless, itcan beturned intoaproperzero-energy solution, identical
to '¥ta2, by replacing w(x) with 1/w(x), but this has the same effect as changing
g to -g, explaining why the new zero-energy solution is the charge conjugate of
'¥ in Eq. (4). Negative chirality zero-energy solutions cannot appear because the
self-duality ofAJ1 implies thatDD= -D~. Therefore suchasolution wouldsatisfy
=
DJ1'¥ 0, which is ruled out.
1 AconstantAo in the Euclidean Weylequationcanalsobeidentifiedwithanon-zerochemical
potential.
3
It is well known that the 't Hooft-Polyakov monopole [2] allows for a nor
malisable chiral zero-energy solution of the Dirac equation [3]. The size of the
core of these non-abelian monopoles is determined by the mass scale set by the
asymptotic value ofthe Higgs field. When the core size shrinks to zero, so does
the support for the zero-mode. Adding to the Higgs field <I> = !<I>aO'a a constant
=
element in D(l), <I>z <I> - 2niz, as it appears in Nahm's work [4], the zero
mode remains normalisable for a finite range ofzdetermined by the Callias index
theorem [5]. Our solution corresponds to zat the boundary of this range, where
the zero-mode fails to be normalisable. This boundary value of zis defined by
=
det(<I>z) 0, and theabelian fieldAo given abovecorresponds to the isospin com
ponent of<I>z responsible for this vanishing eigenvalue. Itdoes imply the support
ofthe zero-mode is no longerconfined to the non-abelian core.
Therehas been anothercontext in which solutions to the Dirac equation in the
background ofa monopole have appeared in the past, namely that ofmonopole
induced proton decay (the Callan-Rubakov effect [6]). Boundary conditions [7]
forthefermions areimposedatthecoreofthemonopole todescribe thescattering
states in the limit where the size ofthe monopole core can be neglected, so as to
properly reflect the breaking of B-L, compatible with the chiral anomaly. There
it is assumed, as for the Jackiw-Rebbi zero-energy solution, that the Higgs field
approaches a non-zero constant at infinity, which through the Yukawa coupling
gives a mass to the fermions2.In ourcase the asymptotic value ofthe Higgs field
(Ao) vanishes and as we will see, the limit of zero monopole core size can be
taken withoutanyapproximation, butattheexpenseofthezero-energy statebeing
non-normalisable.
2. TheAbelian BPSDipole
To find a normalisable zero-energy solution, we have to do something about the
asymptotic behaviour. A natural way to achieve this istoconsider anabelian BPS
(self-dual) dipole, or bipole for short. We now profit from having expressed the
zero-mode in terms ofthefunction w. The bipole field is generated by
logw(x) = 10g(Jxl+x·n) -log(lx+snl+x·n+ s) (8)
x
where the first term represents a monopole at = ()and the second term, with the
opposite sign, an anti-monopole atx= -sn. It isconvenient toexpress w(x) as
_) '1 +'2 -
w(x = S, (9)
'1 +'2+s
2 Foranon-vanishing Higgsmass, thistogetherwiththeDiracmonopolefield, isall thatisleft
when neglecting the core ofthe monopole. In the Bogomol'ny limit considered here, identifying
the Higgsfield withAo (self-dualityimpliedby the BPSequations [8]),the longrange component
ofthe HiggsfieldmodifiestheDiracequation.
