Nuclear Physics B (Proc. Suppl.) 34 (1994) 189-191 North-Holland

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PROCEEDINGS SUPPLEMENTS

Abelian Dominance in Pure Gauge SU(3) Ken Yee a*

~Departmeut of Physics and Astronomy, LSU, Baton Rouge, Louisiana 70803-4001 USA

We give a pedagogical discussion of abelian dominance in QCD and its potential uses. Some of our exploratory SU(3) results in different gauges are presented. Fitting to the Cornell potential we find that in maximal abelian gauge the abelian string tension differs from the non~belian string tension by ,,~ 25%.

This writeup focuses exclusively on abelian dominance. Other aspects of our project are de- scribed in Refs. [1-3].

Based on Higgs models [4] it was speculated that color magnetic monopoles are responsible for QCD confinement. 't Hooft and others [5] sug- gested an abelian projection(AP) scheme for iden- tifying these structures. Upon fixing SU(N) to a gauge with (no more than) residual [U(1)] N-1 local gauge invariance, AP projects SU(N) to a [U ( 1)] N- 1 gauge theory whose N - 1 ab elian fields are originally imbedded inside SU(3). Monopole currents in the [U(1)] N-1 model are readily iden- tified on the lattice according to the famous DeGrand-Toussaint prescription.

The AP is gauge dependent. Performing the AP in a different gauge is equivalent to choosing a different [U(1)] g - 1 imbedding in SU(N). For APSU(2) - - t he U(1) model obtained from the AP of SU(2) - - the authors of Ref. [6] found "strong enhancement of the abelian Wil- son loops and their area-law behavior" in maxi- mal abelian gauge(MA). In particular, they found rough agreement between the abelian and non- abelian string tensions and, further, that the U(1) monopoles are dynamical in the confined phase and relatively static in the finite temperature phase. The latter is consistent with what is ex- pected of confinement-causing monopoles. Com- paring to other gauges, they found that such pos- itive results are obtained only in MA. We have verified independently in SU(3) some of these re- sults: as depicted in Figure 2 our AP string ten-

*internet:kyee@rouge.phys.lsu.edu. The author is sup- ported by U.S. DOE grant DE-FG05-91ER40617.

sion in MA reproduces the nonabelian string ten- sion with "only" a --, 25% discrepancy. Landau and Axial gauge show much larger discrepancies. This unique qualitative "success" of the AP in MA suggests abelian dominance, the notion that APSU(N) in MA tends to capture at long dis- tances the confinement features of SU(N).

Abelian dominance is exciting for two rea- sons. First, since monopole currents are known to be responsible for confinement in abelian mod- els, it lends firepower to the conjecture that monopoles (more precisely, the inverse-AP im- ages of the abelian monopoles) are responsible for nonabelian confinement. Second, it suggests that ultimately an abelian (lattice) gauge model can capture the confinement features of (lattice) QCD. Since [U(1)] u -1 gauge theories are analyt- ically tractable [3], this is encouraging to those seeking an analytical approach to the QCD vac- uum. In the most optimistic scenario, the low en- ergy pure gauge QCD vacuum would be an (ana- lytically tractable) [U(1)] n -1 model ofmonopoles and photons. These structures would interact nonperturbatively with each other, and perturba- lively with the SU(N)/[U(1)] u-1 coset mat ter fields dismissed by AP. The nonperturbative in- teractions would reproduce a la compact QED the familiar dual superconductor ansatz for the QCD vacuum. The coset mat ter fields, presumed heavy, would provide only spectroscopic correc- tions to this dual superconducting abelian vac- Hum.

To proceed with this program, it is essential to have a quantitative idea of how good abelian dom- inance is. Technical details of our AP procedure for SU(3) are given in Appendix A of [1]. As de-

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190 K. Yee/Abelian dominance in pure gauge SU(3)

0.1 ----

2 C~

0.01

0 . 0 0 t

I I I I I I I I i o.15

I

A (3) :< =MA n d a u ~ = A x i a l

- -< / \ x CD ,,, . . . . , "~ 0.10

~ o ~1 \ i / _ 55

4 6 8 L ( lat t ice units)

m.

t

0.00

11 ~ U ~ 0 × 6 = 1 A P S U ( 3 ) , M A : o 6 = O o6=

i

4 6 8 10 12 Rmax/a used for fitting

Figure 1. Nonabelian SU(3) and APSU(3) Creutz ratios in lattice units at fl = 6.0. "a" is the lattice spacing. The APSU(3) Creutz ratios are given in 3 different gauges.

picted in Figure 1, the Creutz ratios of APSU(3) vary with gauge and do not reproduce the non- abelian Creutz ratio. Hence APSU(3) does not reproduce SU(3) values for Wilson loops. What about long-distance observables such as string tension? Tha t the Creutz ratios do not plateau but decrease with increasing Wilson loop size L indicates deviations from perimeter plus area law behavior} Hence string tension cannot be reli- ably read off from Figure 1.

We calculate string tension x by fitting the static quark potential V(R) to Cornell ansatz

V(R) = vo(fl) - e(fl) + ~(fl)R (1) R

If W(R, T) is the expectation value of an R x T Wilson loop, V(R) is given by - T -1 log W(R, T), which should plateau for sufficiently large T. For larger Rs these T-plateau seem to fall outside of our accessible T range. In these instances we stipulate for V(R) its value at T = 12. Fig- ure 2 depicts the result of a simple chisquare fit of V(R) to Eq. (1). The horizontal axis refers to the range R E [1, Rmax] of R values used for

2 We thank T. Suzuki for emphasizing this point to us.

Figure 2. String tension t~ extracted from fits to V(R). The abelian MA ~ differs by -~ 25% from the two horizontal lines, which refer to published 4-1 standard deviation bounds on the fl = 6.0 nonabelian t¢ [7]. t¢(6.0) in Landau gauge is .018(.01) and in Axial gauge is .31(.05). Corre- spondingly the 13 monopole number density in Axial gauge is largest, in Landau gauge smallest.

the fits. 6 E {0,1} refers to two slightly dif- ferent T-plateau choices. As depicted in Fig- ure 2, the AP string tension in MA is stable to variations of Rmax and 6 while the nonabelian t¢ fluctuates wildly. In other words, in contrast to the nonabelian t¢ which cannot be determined by our naive procedure (probably due in part to higher excitations), the AP signal for g is rela- tively clean. Optimists might speculate that this is because the abelian projection flushes out the shorter distance effects making the string ten- sion more transparent. This view comes with the caveat that the AP self-energy and Coulomb co- efficient are both nontrivial: v0(6.0) = .225(.001) and c(6 .0)= .093(.001).

Let us now turn to some speculation about the origin of abelian dominance in MA. Consider the Georgi-Glashow(GG) gauge-Higgs model with an adjoint Higgs field ¢.3 Its relevance to pure gauge

3 O n e c a n extend our discussion to SU(N) using that SU(N) monopoles are imbedded in SU(2) subgroups of

K. Yee/Abelian dominance in pure gauge SU(3) 191

SU(2) is that gauge fixing the latter to MA is equivalent to coupling it to a quenched, frozen Higgs field [8], that is, pure SU(2) fixed to MA is a quenched GG model. In the GG model the electromagnetic field tensor is

A {~aA~)_~ t~aaa'~-~ar~ ~. 0~]~(2)

where ~ is the normalized adjoint Higgs field. fu~ is gauge invariant but the three terms on the RHS of (2) mix under gauge transformations. The evaluation of fu~ is simple in gauges in which one or two of the three terms on the RHS of (2) vanish. For the 't Hooft-Polyakov(TP) magnetic monopole solution [9] in the gauge

(0 u ± i A a u ) ( A ~ ± i A ~ ) = O , ~ a = 6 3 a (3)

fu~ assumes the form of a simple abelian electro- magnetic field tensor,

A 3 3 f ~ = O u ~ - O ~ A , . (4)

In other words, in the gauge (3) one can iden- tify classical TP monopoles by projecting out from the nonabelian fields the A 3 components and treating it as a U(1) gauge field. Therefore, if classical TP monopoles cause confinement, the string tension computed from A 3 abelian Wil- son loops will reproduce the nonabelian string tension in the gauge (3) because in this gauge the abelian Wilson loop correctly measures the monopole electromagnetic field. 4

At this point the Reader will not be surprised to learn that the naive continuum limit of lat- tice MA is indeed (3) and that AP corresponds to working with the Cartan color components of the gauge fields. In this sense, the lattice abelian projection preserves the magnetic field of TP monopoles. By the same logic, AP in other gauges is invalid for identifying TP monopoles be- cause f ~ depends also on A~ '2 in those gauges.

Now three caveats. First, there may be more than one variety of monopoles which contribute to confinement and this argument does not apply

SU(N). 4f~v of (4) incorrec t ly m e a s u r e s m a g n e t i c fields of non- ' t Hoof t -Po lyakov origin. Insofa r as t hese sources do no t c o n t r i b u t e to c o n f i n e m e n t th i s does no t m a t t e r .

to those, which would be incorrectly identified by AP even in MA. Second, quantum fluctuations which our classical argument ignores are expected to modify the TP solutions and invalidate the dis- cussion even for TP monopoles. Third, abelian dominance in the GG model has not been nu- merically established. In view of the recent QCD results, this would clearly be a useful check.

A suggestion [10] is that in QCD A~ plays the role of (I 'a. In the continuum, this ansatz leads to self-dual monopole solutions carrying quantized magnetic and electric charge. It is hard to under- stand how they correspond to the AP monopoles which do not seem to be self-dual. A (perhaps equally unlikely) alternative is that the holes of the lattice play the role of ffa and, hence, univer- sal properties of lattice topology play an essential role in QCD confinement [11].

R E F E R E N C E S

1. M.I. Polikarpov and K. Yee, Phys. Lett. B316 (1993) 333.

2. K. Yee, "Towards an Abelian Formulation of Lattice QCD Confinement," hep-lat/9307019, to appear in Phys. Rev. D.

3. K. Yee, "Compact U(1) x U(1) Model with Minimal Interspecies Interaction," hep- lat/9311003 (November, 1993).

4. A. Polyakov, Phys. Lett. 59B (1975) 82; A. Kronfeld, M. Laursen, G. Schierholz, U. Wiese, Phys. Lett. B198 (1987) 516.

5. S. Mandelstam, Phys. Rept. 23C (1976) 245; G. 't Hooft, Nuel. Phys. B190 (1981) 455; A. Kronfeld, G. Schierholz, U. Wiese, Nucl. Phys. B293 (1987) 461.

6. T. Suzuki and I. Yotsuyanagi, Phys. Rev. D42 (1990) 4257; S. Hioki el. al., Phys. Lett. B272 (1991) 326; T. Suzuki, these Proceedings.

7. For example, G. Bali and K. Schilling, these Proceedings and references therein.

8. K. Yee, Nucl. Phys. B397 (1993) 564. 9. A. Polyakov, JETP Lett. 20, 194 (1974); G.

't Hooft, Nucl. Phys. B79, 276 (1974). 10. J. Smit and A. van der Sijs, Nucl. Phys. B355

(1991) 603. ii. K. Yee, "Inverse Abelian Projection of

Gauss's Law," in preparation.