- Email: [email protected]
Abelian dominance in pure gauge SU (3)
,a r m , m ' .
"~" h'~']l;~l.1,
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 34 (1994) 189-191 North-Holland
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 described 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] suggested an abelian projection(AP) scheme for identifying 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 identified 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 A P S U ( 2 ) - - t h e U(1) model obtained from the AP of S U ( 2 ) - - t h e authors of Ref. [6] found "strong enhancement of the abelian Wilson loops and their area-law behavior" in maximal abelian gauge(MA). In particular, they found rough agreement between the abelian and nonabelian 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 expected of confinement-causing monopoles. Comparing to other gauges, they found that such positive results are obtained only in MA. We have verified independently in SU(3) some of these results: as depicted in Figure 2 our AP string ten*internet:[email protected]. The author is supported by U.S. DOE grant DE-FG05-91ER40617.
sion in MA reproduces the nonabelian string tension 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 distances the confinement features of SU(N). Abelian dominance is exciting for two reasons. First, since monopole currents are known to be responsible for confinement in abelian models, it lends firepower to the conjecture that monopoles (more precisely, the inverse-AP images 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 analytically tractable [3], this is encouraging to those seeking an analytical approach to the QCD vacuum. In the most optimistic scenario, the low energy pure gauge QCD vacuum would be an (analytically tractable) [U(1)] n - 1 model ofmonopoles and photons. These structures would interact nonperturbatively with each other, and perturbalively with the SU(N)/[U(1)] u-1 coset matter fields dismissed by AP. The nonperturbative interactions would reproduce a la compact QED the familiar dual superconductor ansatz for the QCD vacuum. The coset matter fields, presumed heavy, would provide only spectroscopic corrections to this dual superconducting abelian vacHum.
To proceed with this program, it is essential to have a quantitative idea of how good abelian dominance is. Technical details of our AP procedure for SU(3) are given in Appendix A of [1]. As de-
0920-5632/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSD10920-5632(94)00238-Q
K. Yee/Abeliandominance in pure gauge SU(3)
190
I
I I I I
I I I i
o.15
I A
0.1
---,,,
--
. ~1\
( 3 ) : < =MA
.
.
ndau
~ U ~
~=Axial
,
.
i/
_
0 ×6=1
APSU(3),MA:o6=O
o6=
11
CD
"~ 0.10 ~o 55
i
2 C~
0.01 m.
t
0.00
0.00t
4 6 L (lattice units)
8
6
Rmax/a
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 nonabelian Creutz ratio. Hence APSU(3) does not reproduce SU(3) values for Wilson loops. What about long-distance observables such as string tension? T h a 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 reliably 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 R
4
(1)
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. Figure 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
12
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). Correspondingly the 13 monopole number density in Axial gauge is largest, in Landau gauge smallest.
the fits. 6 E {0,1} refers to two slightly different T-plateau choices. As depicted in Figure 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 relatively clean. Optimists might speculate that this is because the abelian projection flushes out the shorter distance effects making the string tension more transparent. This view comes with the caveat that the AP self-energy and Coulomb coefficient 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 extend our discussion t o SU(N) using that SU(N) monopoles are imbedded in SU(2) subgroups of
3One can
2We thank T. Suzuki for emphasizing this point to us.
8 10 used for fitting
K. Yee/Abelian dominance in pure gauge SU(3)
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=63a
(3)
fu~ assumes the form of a simple abelian electromagnetic field tensor, f ~ = O u A 3~ - O ~ A ,3.
(4)
In other words, in the gauge (3) one can identify 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 Wilson 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 lattice 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 because 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) i n c o r r e c t l y m e a s u r e s m a g n e t i c fields of n o n 't H o o f t - P o l y a k o v origin. I n s o f a r as t h e s e s o u r c e s do n o t c o n t r i b u t e to c o n f i n e m e n t t h i s d o e s n o t m a t t e r .
191
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 discussion even for TP monopoles. Third, abelian dominance in the GG model has not been numerically 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 understand 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, universal properties of lattice topology play an essential role in QCD confinement [11]. REFERENCES
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," heplat/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.
"~" h'~']l;~l.1,
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 34 (1994) 189-191 North-Holland
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 described 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] suggested an abelian projection(AP) scheme for identifying 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 identified 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 A P S U ( 2 ) - - t h e U(1) model obtained from the AP of S U ( 2 ) - - t h e authors of Ref. [6] found "strong enhancement of the abelian Wilson loops and their area-law behavior" in maximal abelian gauge(MA). In particular, they found rough agreement between the abelian and nonabelian 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 expected of confinement-causing monopoles. Comparing to other gauges, they found that such positive results are obtained only in MA. We have verified independently in SU(3) some of these results: as depicted in Figure 2 our AP string ten*internet:[email protected]. The author is supported by U.S. DOE grant DE-FG05-91ER40617.
sion in MA reproduces the nonabelian string tension 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 distances the confinement features of SU(N). Abelian dominance is exciting for two reasons. First, since monopole currents are known to be responsible for confinement in abelian models, it lends firepower to the conjecture that monopoles (more precisely, the inverse-AP images 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 analytically tractable [3], this is encouraging to those seeking an analytical approach to the QCD vacuum. In the most optimistic scenario, the low energy pure gauge QCD vacuum would be an (analytically tractable) [U(1)] n - 1 model ofmonopoles and photons. These structures would interact nonperturbatively with each other, and perturbalively with the SU(N)/[U(1)] u-1 coset matter fields dismissed by AP. The nonperturbative interactions would reproduce a la compact QED the familiar dual superconductor ansatz for the QCD vacuum. The coset matter fields, presumed heavy, would provide only spectroscopic corrections to this dual superconducting abelian vacHum.
To proceed with this program, it is essential to have a quantitative idea of how good abelian dominance is. Technical details of our AP procedure for SU(3) are given in Appendix A of [1]. As de-
0920-5632/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSD10920-5632(94)00238-Q
K. Yee/Abeliandominance in pure gauge SU(3)
190
I
I I I I
I I I i
o.15
I A
0.1
---,,,
--
. ~1\
( 3 ) : < =MA
.
.
ndau
~ U ~
~=Axial
,
.
i/
_
0 ×6=1
APSU(3),MA:o6=O
o6=
11
CD
"~ 0.10 ~o 55
i
2 C~
0.01 m.
t
0.00
0.00t
4 6 L (lattice units)
8
6
Rmax/a
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 nonabelian Creutz ratio. Hence APSU(3) does not reproduce SU(3) values for Wilson loops. What about long-distance observables such as string tension? T h a 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 reliably 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 R
4
(1)
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. Figure 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
12
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). Correspondingly the 13 monopole number density in Axial gauge is largest, in Landau gauge smallest.
the fits. 6 E {0,1} refers to two slightly different T-plateau choices. As depicted in Figure 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 relatively clean. Optimists might speculate that this is because the abelian projection flushes out the shorter distance effects making the string tension more transparent. This view comes with the caveat that the AP self-energy and Coulomb coefficient 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 extend our discussion t o SU(N) using that SU(N) monopoles are imbedded in SU(2) subgroups of
3One can
2We thank T. Suzuki for emphasizing this point to us.
8 10 used for fitting
K. Yee/Abelian dominance in pure gauge SU(3)
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=63a
(3)
fu~ assumes the form of a simple abelian electromagnetic field tensor, f ~ = O u A 3~ - O ~ A ,3.
(4)
In other words, in the gauge (3) one can identify 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 Wilson 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 lattice 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 because 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) i n c o r r e c t l y m e a s u r e s m a g n e t i c fields of n o n 't H o o f t - P o l y a k o v origin. I n s o f a r as t h e s e s o u r c e s do n o t c o n t r i b u t e to c o n f i n e m e n t t h i s d o e s n o t m a t t e r .
191
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 discussion even for TP monopoles. Third, abelian dominance in the GG model has not been numerically 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 understand 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, universal properties of lattice topology play an essential role in QCD confinement [11]. REFERENCES
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," heplat/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.