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leads with V12 ∝ (φ1 φ2 )2 to a system of equations for the functions f (ρ) and
±(ρ) which is almost identical to the coupled system of equations (26) and(27)
for the abelian vortex. As for the Nielsen“Olesen vortex or the ™t Hooft“Polyakov
monopole, the topological stability of this vortex is ultimately guaranteed by the
non-vanishing values of the Higgs ¬elds, enforced by the self-interactions and the
asymptotic alignment of gauge and Higgs ¬elds. This stability manifests itself in
the quantization of the magnetic ¬‚ux (cf.(125))

B · dσ = ’
m= . (195)

In this generalized Higgs model, ¬elds can be classi¬ed according to their mag-
netic ¬‚ux, which either vanishes as for the zero energy con¬gurations or takes
on the value (195). With this classi¬cation, one can associate a Z2 symmetry
74 F. Lenz

similar to the center symmetry with singular gauge transformations connecting
the two classes. Unlike center re¬‚ections (181), singular gauge transformations
change the value of the action. It has been argued [80] that, within the 2+1
dimensional Higgs model, this “topological symmetry” is spontaneously bro-
ken with the vacuum developing a domain structure giving rise to con¬nement.
Whether this happens is a dynamical issue as complicated as the formation
of ¬‚ux tubes in Type II superconductors discussed on p.18. This spontaneous
symmetry breakdown requires the center vortices to condense as a result of an
attractive vortex“vortex interaction which makes the square of the vortex mass
zero or negative. Extensions of such a scenario to pure gauge theories in 3+1
dimensions have been suggested [81,82].

8.5 The Spectrum of the SU(2) Yang“Mills Theory

Based on the results of Sect. 8.3 concerning the symmetry and topology of Yang“
Mills theories at ¬nite extension, I will deduce properties of the spectrum of the
SU(2) Yang“Mills theory in the con¬ned, center-symmetric phase.

• In the center-symmetric phase,

Z|0 = |0 ,

the vacuum expectation value of the Polyakov loop vanishes (188).
• The correlation function of Polyakov loops yields the interaction energy V
of static color charges (in the fundamental representation)
exp {’LV (r)} = 0|T tr P xE tr P (0) |0 , r2 = xE . (196)
⊥ ⊥

• Due to the rotational invariance in Euclidean space, xE can be chosen to

point in the time direction. After insertion of a complete set of excited states

| n’ |tr P (0) |0 | e’En’ r .
exp {’LV (r)} = (197)

In the con¬ned phase, the ground state does not contribute (188). Since
P xE is odd under re¬‚ections only odd excited states,

Z|n’ = ’|n’ ,

contribute to the above sum. If the spectrum exhibits a gap,

En’ ≥ E1’ > 0,

the potential energy V increases linearly with r for large separations,

V (r) ≈ for r ’ ∞
r and L > Lc . (198)
Topological Concepts in Gauge Theories 75

• The linear rise with the separation, r, of two static charges (cf. (189)) is a
consequence of covariance and the existence of a gap in the states excited by
the Polyakov-loop operator. The slope of the con¬ning potential is the string
tension σ. Thus, in Yang“Mills theory at ¬nite extension, the phenomenon
of con¬nement is connected to the presence of a gap in the spectrum of
Z’odd states,
E’ ≥ σL , (199)
which increases linearly with the extension of the compact direction. When
applied to the vacuum, the Polyakov-loop operator generates states which
contain a gauge string winding around the compact direction. The lower
limit (199) is nothing else than the minimal energy necessary to create such
a gauge string in the con¬ning phase. Two such gauge strings, unlike one, are
not protected topologically from decaying into the ground state or Z = 1
excited states. We conclude that the states in the Z = ’1 sector contain
Z2 - stringlike excitations with excitation energies given by σL. As we have
seen, at the classical level, gauge ¬elds with vanishing action exist which
wind around the compact direction. Quantum mechanics lifts the vacuum
degeneracy and assigns to the corresponding states the energy (199).
• Z’even operators in general will have non-vanishing vacuum expectation
values and such operators are expected to generate the hadronic states with
the gap determined by the lowest glueball mass E+ = mgb for su¬ciently
large extension mgb L 1.
• SU(2) Yang“Mills theory contains two sectors of excitations which, in the
con¬ned phase, are not connected by any physical process.
“ The hadronic sector, the sector of Z’even states with a mass gap (ob-
tained from lattice calculations) E+ = mgb ≈ 1.5 GeV
“ The gluonic sector, the sector of Z’odd states with mass gap E’ = σL.
• When compressing the system, the gap in the Z = ’1 sector decreases
to about 650 MeV at Lc ≈ 0.75fm, (Tc ≈ 270 MeV). According to SU(3)
lattice gauge calculations, when approaching the critical temperature Tc ≈
220 MeV, the lowest glueball mass decreases. The extent of this decrease is
controversial. The value mgb (Tc ) = 770 MeV has been determined in [83,84]
while in a more recent calculation [85] the signi¬cantly higher value of
1250 MeV is obtained for the glueball mass at Tc .
• In the decon¬ned or plasma phase, the center symmetry is broken. The ex-
pectation value of the Polyakov loop does not vanish. Debye screening of the
fundamental charges takes place and formation of ¬‚ux tubes is suppressed.
Although the decon¬ned phase has been subject of numerous numerical in-
vestigations, some conceptual issues remain to be clari¬ed. In particular, the
origin of the exceptional realization of the center symmetry is not under-
stood. Unlike symmetries of nearly all other systems in physics, the center
symmetry is realized in the low temperature phase and broken in the high
temperature phase. The con¬nement“decon¬nement transition shares this
exceptional behavior with the “inverse melting” process which has been ob-
served in a polymeric system [86] and in a vortex lattice in high-Tc super-
conductors [87]. In the vortex lattice, the (inverse) melting into a crystalline
76 F. Lenz

state happens as a consequence of the increase in free energy with increasing
disorder which, in turn, under special conditions, may favor formation of a
vortex lattice. Since nature does not seem to o¬er a variety of possibilities for
inverse melting, one might guess that a similar mechanism is at work in the
con¬nement“decon¬nement transition. A solution of this type would be pro-
vided if the model of broken topological Z2 symmetry discussed in Sect. 8.4
could be substantiated. In this model the con¬nement“decon¬nement tran-
sition is driven by the dynamics of the “disorder parameter” [80] which
exhibits the standard pattern of spontaneous symmetry breakdown.
The mechanism driving the con¬nement“decon¬nement transition must also
be responsible for the disparity in the energies involved. As we have seen,
glueball masses are of the order of 1.5 GeV. On the other hand, the maximum
in the spectrum of the black-body radiation increases with temperature and
reaches according to Planck™s law at T = 220 MeV a value of 620 MeV.
A priori one would not expect a dissociation of the glueballs at such low
temperatures. According to the above results concerning the Z = ±1 sectors,
the phase transition may be initiated by the gain in entropy through coupling
of the two sectors which results in a breakdown of the center symmetry. In
this case the relevant energy scale is not the glueball mass but the mass gap
of the Z = ’1 states which, at the extension corresponding to 220 MeV,
coincides with the peak in the energy density of the blackbody-radiation.

9 QCD in Axial Gauge

In close analogy to the discussion of the various ¬eld theoretical models which
exhibit topologically non-trivial excitations, I have described so far SU (2) Yang“
Mills theory from a rather general point of view. The combination of symmetry
and topological considerations and the assumption of a con¬ning phase has led
to intriguing conclusions about the spectrum of this theory. To prepare for more
detailed investigations, the process of elimination of redundant variables has
to be carried out. In order to make the residual gauge symmetry (the center
symmetry) manifest, the gauge condition has to be chosen appropriately. In
most of the standard gauges, the center symmetry is hidden and will become
apparent in the spectrum only after a complete solution. It is very unlikely
that approximations will preserve the center symmetry as we have noticed in
the context of the perturbative evaluation of the e¬ective potential in QED (cf.
(178) and (179)). Here I will describe SU (2) Yang“Mills theory in the framework
of axial gauge, in which the center re¬‚ections can be explicitly constructed and
approximation schemes can be developed which preserve the center symmetry.

9.1 Gauge Fixing

We now carry out the elimination of redundant variables and attempt to elim-
inate the 3-component of the gauge ¬eld A3 (x). Formally this can be achieved
Topological Concepts in Gauge Theories 77

by applying the gauge transformation
„¦(x) = P exp ig dz A3 (x⊥ , z) .

It is straightforward to verify that the gauge transformed 3-component of the
gauge ¬eld indeed vanishes (cf. (90))

‚3 „¦ † (x) = 0.
A3 (x) ’ „¦ (x) A3 (x) +

However, this gauge transformation to axial gauge is not quite legitimate. The
gauge transformation is not periodic

„¦(x⊥ , x3 + L) = „¦(x⊥ , x3 ).

In general, gauge ¬elds then do not remain periodic either under transformation
with „¦. Furthermore, with A3 also the gauge invariant trace of the Polyakov
loop (182) is incorrectly eliminated by „¦. These shortcomings can be cured, i.e.
periodicity can be preserved and the loop variables tr P (x⊥ ) can be restored
with the following modi¬ed gauge transformation
x3 /L
„¦ag (x) = „¦D (x⊥ ) P † (x⊥ ) „¦(x) . (200)

The gauge ¬xing to axial gauge thus proceeds in three steps
• Elimination of the 3-component of the gauge ¬eld A3 (x)
• Restoration of the Polyakov loops P (x⊥ )
• Elimination of the gauge variant components of the Polyakov loops P (x⊥ )
by diagonalization

„¦D (x⊥ ) P (x⊥ )„¦D (x⊥ ) = eigLa3 (x⊥ ) „ /2
. (201)

Generating Functional. With the above explicit construction of the appro-
priate gauge transformations, we have established that the 3-component of the
gauge ¬eld indeed can be eliminated in favor of a diagonal x3 -independent ¬eld
a3 (x⊥ ). In the language of the Faddeev“Popov procedure, the axial gauge con-
dition (cf. (132)) therefore reads

π „3
f [A] = A3 ’ a3 + . (202)
gL 2
The ¬eld a3 (x⊥ ) is compact,
π π
’ ¤ a3 (x⊥ ) ¤
a3 = a3 (x⊥ ), .
gL gL
It is interesting to compare QED and QCD in axial gauge in order to identify
already at this level properties which are related to the non-abelian character of
78 F. Lenz

QCD. In QED the same procedure can be carried out with omission of the third
step. Once more, a lower dimensional ¬eld has to be kept for periodicity and
gauge invariance. However, in QED the integer part of a3 (x⊥ ) cannot be gauged
away; as winding number of the mapping S 1 ’ S 1 it is protected topologically. In
QCD, the appearance of the compact variable is ultimately due to the elimination
of the gauge ¬eld A3 , an element of the Lie algebra, in favor of P (x⊥ ), an element
of the compact Lie group.
With the help of the auxiliary ¬eld a3 (x⊥ ), the generating functional for
QCD in axial gauge is written as

π „3 d4 xJ µ Aµ
d[a3 ]d [A] ∆f [A] δ A3 ’ a3 + eiS[A]+i
Z [J] = . (203)
gL 2

This generating functional contains as dynamical variables the ¬elds a3 (x⊥ ),
A⊥ (x) with
A⊥ (x) = {A0 (x), A1 (x), A2 (x)}.
It is one of the unique features of axial gauge QCD that the Faddeev“Popov
determinant (cf. (136) and 135))

∆f [A] = | det D3 |

can be evaluated in closed form
det D3 1
cos2 gLa3 (x⊥ )/2 ,
(det ‚3 )3 2

and absorbed into the measure

d4 xJA
D[a3 ]d [A⊥ ] eiS [A⊥ ,a3 ’ gL ]+i
Z [J] = .

The measure

cos2 (gLa3 (x⊥ )/2) ˜ a3 (x⊥ )2 ’ (π/gL)2 da3 (x⊥ )
D [a3 ] = (204)


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