ńņš. 17 |

leads with V12 ā (Ļ1 Ļ2 )2 to a system of equations for the functions f (Ļ) and

Ī±(Ļ) which is almost identical to the coupled system of equations (26) and(27)

for the abelian vortex. As for the Nielsenā“Olesen vortex or the ā™t Hooftā“Polyakov

monopole, the topological stability of this vortex is ultimately guaranteed by the

non-vanishing values of the Higgs ļ¬elds, enforced by the self-interactions and the

asymptotic alignment of gauge and Higgs ļ¬elds. This stability manifests itself in

the quantization of the magnetic ļ¬‚ux (cf.(125))

2Ļ

B Ā· dĻ = ā’

m= . (195)

g

S2

In this generalized Higgs model, ļ¬elds can be classiļ¬ed according to their mag-

netic ļ¬‚ux, which either vanishes as for the zero energy conļ¬gurations or takes

on the value (195). With this classiļ¬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 reļ¬‚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 ātopological symmetryā is spontaneously bro-

ken with the vacuum developing a domain structure giving rise to conļ¬nement.

Whether this happens is a dynamical issue as complicated as the formation

of ļ¬‚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 vortexā“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) Yangā“Mills Theory

Based on the results of Sect. 8.3 concerning the symmetry and topology of Yangā“

Mills theories at ļ¬nite extension, I will deduce properties of the spectrum of the

SU(2) Yangā“Mills theory in the conļ¬ned, center-symmetric phase.

ā¢ In the center-symmetric phase,

Z|0 = |0 ,

the vacuum expectation value of the Polyakov loop vanishes (188).

ā¢ The correlation function of Polyakov loops yields the interaction energy V

of static color charges (in the fundamental representation)

2

exp {ā’LV (r)} = 0|T tr P xE tr P (0) |0 , r2 = xE . (196)

ā„ ā„

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

ā„

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

| nā’ |tr P (0) |0 | eā’Enā’ r .

2

exp {ā’LV (r)} = (197)

nā’

In the conļ¬ned phase, the ground state does not contribute (188). Since

P xE is odd under reļ¬‚ections only odd excited states,

ā„

Z|nā’ = ā’|nā’ ,

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

Enā’ ā„ E1ā’ > 0,

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

E1

V (r) ā for r ā’ ā

r and L > Lc . (198)

L

Topological Concepts in Gauge Theories 75

ā¢ 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 conļ¬ning potential is the string

tension Ļ. Thus, in Yangā“Mills theory at ļ¬nite extension, the phenomenon

of conļ¬nement is connected to the presence of a gap in the spectrum of

Zā’odd states,

Eā’ ā„ ĻL , (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 conļ¬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 = ā’1 sector contain

Z2 - stringlike excitations with excitation energies given by ĻL. As we have

seen, at the classical level, gauge ļ¬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).

ā¢ Zā’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 suļ¬ciently

large extension mgb L 1.

ā¢ SU(2) Yangā“Mills theory contains two sectors of excitations which, in the

conļ¬ned phase, are not connected by any physical process.

ā“ The hadronic sector, the sector of Zā’even states with a mass gap (ob-

tained from lattice calculations) E+ = mgb ā 1.5 GeV

ā“ The gluonic sector, the sector of Zā’odd states with mass gap Eā’ = ĻL.

ā¢ When compressing the system, the gap in the Z = ā’1 sector decreases

to about 650 MeV at Lc ā 0.75fm, (Tc ā 270 MeV). According to SU(3)

lattice gauge calculations, when approaching the critical temperature Tc ā

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 signiļ¬cantly higher value of

1250 MeV is obtained for the glueball mass at Tc .

ā¢ In the deconļ¬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 ļ¬‚ux tubes is suppressed.

Although the deconļ¬ned phase has been subject of numerous numerical in-

vestigations, some conceptual issues remain to be clariļ¬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 conļ¬nementā“deconļ¬nement transition shares this

exceptional behavior with the āinverse meltingā 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 oļ¬er a variety of possibilities for

inverse melting, one might guess that a similar mechanism is at work in the

conļ¬nementā“deconļ¬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 conļ¬nementā“deconļ¬nement tran-

sition is driven by the dynamics of the ādisorder parameterā [80] which

exhibits the standard pattern of spontaneous symmetry breakdown.

The mechanism driving the conļ¬nementā“deconļ¬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 Planckā™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 = Ā±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 = ā’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 ļ¬eld theoretical models which

exhibit topologically non-trivial excitations, I have described so far SU (2) Yangā“

Mills theory from a rather general point of view. The combination of symmetry

and topological considerations and the assumption of a conļ¬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 eļ¬ective potential in QED (cf.

(178) and (179)). Here I will describe SU (2) Yangā“Mills theory in the framework

of axial gauge, in which the center reļ¬‚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 ļ¬eld A3 (x). Formally this can be achieved

Topological Concepts in Gauge Theories 77

by applying the gauge transformation

x3

ā„¦(x) = P exp ig dz A3 (xā„ , z) .

0

It is straightforward to verify that the gauge transformed 3-component of the

gauge ļ¬eld indeed vanishes (cf. (90))

1

ā‚3 ā„¦ ā (x) = 0.

A3 (x) ā’ ā„¦ (x) A3 (x) +

ig

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

gauge transformation is not periodic

ā„¦(xā„ , x3 + L) = ā„¦(xā„ , x3 ).

In general, gauge ļ¬elds then do not remain periodic either under transformation

with ā„¦. Furthermore, with A3 also the gauge invariant trace of the Polyakov

loop (182) is incorrectly eliminated by ā„¦. These shortcomings can be cured, i.e.

periodicity can be preserved and the loop variables tr P (xā„ ) can be restored

with the following modiļ¬ed gauge transformation

x3 /L

ā„¦ag (x) = ā„¦D (xā„ ) P ā (xā„ ) ā„¦(x) . (200)

The gauge ļ¬xing to axial gauge thus proceeds in three steps

ā¢ Elimination of the 3-component of the gauge ļ¬eld A3 (x)

ā¢ Restoration of the Polyakov loops P (xā„ )

ā¢ Elimination of the gauge variant components of the Polyakov loops P (xā„ )

by diagonalization

3

ā

ā„¦D (xā„ ) P (xā„ )ā„¦D (xā„ ) = eigLa3 (xā„ ) Ļ„ /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 ļ¬eld indeed can be eliminated in favor of a diagonal x3 -independent ļ¬eld

a3 (xā„ ). In the language of the Faddeevā“Popov procedure, the axial gauge con-

dition (cf. (132)) therefore reads

Ļ Ļ„3

f [A] = A3 ā’ a3 + . (202)

gL 2

The ļ¬eld a3 (xā„ ) is compact,

Ļ Ļ

ā’ ā¤ a3 (xā„ ) ā¤

a3 = a3 (xā„ ), .

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 ļ¬eld has to be kept for periodicity and

gauge invariance. However, in QED the integer part of a3 (xā„ ) cannot be gauged

away; as winding number of the mapping S 1 ā’ S 1 it is protected topologically. In

QCD, the appearance of the compact variable is ultimately due to the elimination

of the gauge ļ¬eld A3 , an element of the Lie algebra, in favor of P (xā„ ), an element

of the compact Lie group.

With the help of the auxiliary ļ¬eld a3 (xā„ ), the generating functional for

QCD in axial gauge is written as

Ļ Ļ„3 d4 xJ Āµ AĀµ

d[a3 ]d [A] āf [A] Ī“ A3 ā’ a3 + eiS[A]+i

Z [J] = . (203)

gL 2

This generating functional contains as dynamical variables the ļ¬elds a3 (xā„ ),

Aā„ (x) with

Aā„ (x) = {A0 (x), A1 (x), A2 (x)}.

It is one of the unique features of axial gauge QCD that the Faddeevā“Popov

determinant (cf. (136) and 135))

āf [A] = | det D3 |

can be evaluated in closed form

det D3 1

cos2 gLa3 (xā„ )/2 ,

=

(det ā‚3 )3 2

L

xā„

and absorbed into the measure

d4 xJA

Ļ

D[a3 ]d [Aā„ ] eiS [Aā„ ,a3 ā’ gL ]+i

Z [J] = .

The measure

cos2 (gLa3 (xā„ )/2) Ī˜ a3 (xā„ )2 ā’ (Ļ/gL)2 da3 (xā„ )

D [a3 ] = (204)

ńņš. 17 |