U(3) four-dimensional lattice gauge theory and Monte Carlo calculations

Nuclear Physxcs B210[FS6] (1982) 377-387 © North-Holland Pubhslung Company

U(3) F O U R - D I M E N S I O N A L LA'ITICE GAUGE THEORY AND M O N T E CARLO CALCULATIONS Machael C R E U T Z

Brookhaven National Laboratory. Upton, New York 11973. USA K J M MORIARTY

Deutsches Elektronen-Synchrotron DES Y, llamburg, W Germany and Department of Mathemattc**, Royal Hollowav College. Englefield Green, Surrey, TW20 OEX, UK Received 8 June 1982

Monte Carlo results for the pure U(3) lattice gauge theory on a 64 lattice are reported. Wdson loops and the stnng tensaon are presented The first-order phase transmon an U(3) Is reflected qmte clearly an a dlscontmmt~ m the stnng tensaon at/~ = tic The U(I) factor of U(3) ~s extracted using the deternunant of the Wdson loops As expected, the U(I) component appears to deconfme at the phase transtt~on

In a previous paper [1], we found a first-order phase transition in pure U(3) gauge theory on a four-dimensional space-time lattice. In order to examine the nature of this transition in more detail, we study the Wilson loops and, hence, the string tension for this gauge group. The U(3) gauge group contains SU(3) and U(1) components which should decouple at low temperature. We should then be left with the confinement of SU(3) color and the deconfinement of U(I) charges. In a recent paper [2] we verified the analogous behavior with the gauge group U(2). This paper extends these results to U(3). We consider U ( N ) latuce gauge theory in four dimensions. The lattice spacing ts a cutoff for small distances and, hence, the inverse lattice spacing is an ultraviolet cutoff m momentum space. A matrix U,: ~ U( N ) is associated with each link joining the nearest-neighbor lattice sttes, denoted by t and j. We decompose these matrices in the form

U,:=exp(tO,:)~:, where U,: is an N × N unitary unimodular matrix of S U ( N ) and 0 is an angle * Permanent address 377

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M Creutz, K J M Mortarty / U(3)four-dtmenslonallattt~e gauge theot3

associated with the compact U(I) gauge degree of freedom. We cover the U(N) gauge group manifold when the angle/9 covers the interval [0, 2~r/N) and U,,j covers SU(N). By traversing a link in the reverse direction, we get the inverse group element, i.e., Uj, = (U,/) The expectauon value of the observable

(A(fl))

J

A[U] is defined by

1 Z(fl)f(o~j>dU'J)

A[U]exp(-BS[U])"

where Z(fl) is the partiuon function defined to normalize the expectation value of the operator 1. In the above expression, fl is the inverse temperature which is related to the bare coupling constant by fl = 2N/gg and the measure is the normalized invariant Haar measure for the U(N) gauge group. We define the action S as a sum over all unoriented plaquettes [] of the lattice S=ZS o=E

(l

I-~ReTrU

)

o ,

when Up = U,jU~kUklUa is the parallel transporter around the plaquette [] with the boundary ij, jk, kl and h. We use periodic boundary conditions throughout our calculations and the standard Monte Carlo method of Metropolis et al. [3, 4] to equilibrate our lattice. From now on we will speoalize to the case of N = 3, i.e. U(3). The Wilson loop [5] on the lattice is defined by the expectation value

W(I, J ) = ½(ReTr Uc), where C is a rectangular contour of dimensions I and J and the product of link variables around the contour C is denoted by Uc. The leading-order high-temperature expansion for the Wilson loop is

W(I, J) = (~fl)ts × (1 + O ( f l : ) ) ,

(l)

while the leading-order low-temperature expansion for the average action per plaquette is [6] ( E ) = l - W ( l , 1) =

+ o(,-

2).

(2)

For rectangular dimensions large compared to the correlation length, the Wilson

M Creutz, K J ~4 Mormrt) / U(3)four-dlmemtonallamcegauge theory

379

loop should assume the asymptotic form W ( I , J) = exp(-A - B. I J - C. 2 ( 1 + J ) ) . where, for convenience, we have set the lattice spacing to 1, and for a gtven/3, the parameters A, B and C are constants. When the asymptotic form of the above equation apphes, the string tension B is easily extracted by calculating the quantity

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F~g 2 The evolution of the average acuon per plaquette ( E ) for pure U(3) gauge theory on a 64 lattice as a functaon of the n u m b e r of tterataons through the lattxce for nuxed phase starlang lattaces for various values of the reverse temperature #

380

M Creutz, K J M Monarty / U(3) four-dtmenstonal lattwe gauge theoO,

The leading-order high-temperature expansion for the string tension is

X(1, J ) = -in(~/3)

(3)

"F- 0 ( / ~ 2 ) .

We can extract the U(I) component of U(3) by calculating the deternunant [8] for each loop considered as a 3 × 3 matrix in U(3) and averaging over all similar loops in each configuration to give the average deterrmnants denoted by

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Fig 3 The Wilson loops W(I, J) for pure U(3) gauge theory on a 64 lattice as a function of the reverse temperature B- The full upward triangles represent (I, J ) = (1, 1), the full circles (2, 1), the crosses (3, I), the open circles (2, 2), the full downward triangles (3, 2) and the full squares (3, 3) The curves represent the leading-order hlgh-temperature expansion of eq (1)

M Creutz, K J M Mortarty / U(3) four-dtmenstonal latttce gauge theor~

381

F r o m these quanuties we form the new logarithmic ratto

[W(I, J)~'(I-

1, J - 1) l _W( l , j ~ ~) -~( i - - ~ f ~) ] .

~(l,J)=-in

If at weak coupling the theory does not confine U(I) charges, then as the loop grows this quantity should go to zero in the low-temperature region. The leading-order high-temperature expansion for the average determinants is

W(I, J)=

[_~/~3 + 0(/~5)] 'g,

(4)

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Fig 4 (a,b) The funcuon X(I, J) for pure U(3) gauge theory on a 64 lattace as a funcuon of the reverse temperature B The full upward tnangles represent (I, J ) = (I, I), the full circles (2,2), the crosses (3,2) and the open circles (3,3) Also shown m the diagram is the leading-order hagh-temperature expansion of eq (3)

382

M Creutz, K J M Mormrty / U(3)four-dtmemtonal latttce gauge theory

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Fig 4 (continued) and for the quantity

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(5)

In fig. 1 we show the average action per plaquette for pure U(3) gauge theory as a function of the inverse temperature fl on a 64 lattice. To obtain these results, we first performed 200 iterations through the lattice with 20 Monte Carlo updates per hnk. This appears to adequately equilibrate the lattice. We then averaged over the next 100 iterations through the lattice. We used disordered starts for fl ~< 5.5, mixed phase [9] starts for 5.6 < fl < 9.0 and ordered starts for fl >/9.0. Mixed phase runs were used in the crossover region in order to avoid problems of supercoohng or superheatmg assocmted with disordered and ordered starts, respecuvely. In the mixed phase starts, the fourth axis of the euclidean lattice, the time axis, was dwided in two w~th the upper half of the link variables disordered and the lower half ordered. In fig. 1 the leading-order high- and low-temperature expansions of eqs. (!) and (2), respectively, are also shown. In fig. 2 we show some mixed phase runs for the average action per plaquette in the vicinity of the cntical reverse temperature and we see evidence for a first-order phase transition at the reverse temperature of fl~ = 7.06 + 0.01. The Wilson loops of size up to 3 × 3 are presented m fig. 3. Eq. (1) accurately describes the strong coupling behavior.

M Creutz, K J M Morzarty / U(3)four-dtmert~zonal latttce gauge theol."

383

In fig. 4a we present the quantity X(I, J), for (I, J ) = (1, 1), (2,2), (3,2) and (3,3) as a function of the inverse temperature/3. Also shown in fig. 4 is the high-temperature expansion of eq. (3) which agrees well with the Monte Carlo data for/3 < 7. In fig. 4b we show the vicinity of the critical inverse temperature m some detail to exhibit the sharp jump, presumably a discontinuity, m the string tension. At large fl, the U(1) part of our matrices should decouple and leave an effecuve SU(3) theory. In fig. 5 we present the X ratios for both U(3) and SU(3) gauge theortes. The U(3) results mimic SU(3) results shifted by approximately l.l units in /3. This shift may be calculable perturbatwely. In fig. 6 we show the average U(l) acuon per plaquette ( E ) = l - W(l, l) as a function of the inverse temperature fl on a 6 4 lattice. The leading-order high-temper-

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M Creutz, K J. M Monarty / U(3)four-dlmenstonal latnce gauge theo~"

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M Creutz, K J M Monarty/ U(3)four-dtmenswnallattwegauge theory

385

ature expansion of eq. (4) is also shown. In fig. 7 some mixed phase runs for (bS) near the criucal inverse temperature are shown. The first-order nature of the transition is also quite clear in this quantity; this contrasts with the second-order transition in the pure U(I) model [10]. The U ( I ) Wilson loops W ( I , J ) of size up to 3 × 3 are shown in fig. 8 with eq. (4) shown for comparison. In fig. 9 we show the logarithmic ratios 2(1, J ) for (I, J ) = (1, 1), (2,2), (3,2) and (3, 3) as a function of the reverse temperature ft. We also indicate the leading-order high-temperature expansion of eq. (5). In the low-temperature region, these quantities decrease with loop size faster than the quantities X(I, J ) shown m fig. 4. Indeed, the expectation is that the quantities 2(1, J ) should go to zero in the low-temperature region as the loop size increases. Thus, the determmant of the loops would appear not to confine U(I) charges in the low-temperature region as in the pure U(I) gauge theory [10, 1 1].

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F~g 8 The Wdson loops ~'(I, J) for the U(I) component of pure U(3) gauge theory,on a 6 4 latuce as a function of the ,nverse temperature/J The full upward triangles represent (I, J ) = (I, 1), the full c~rcles (2, 1), the crosses (3, I) and the open c~rcles(2, 2) The curves represent the leading-orderh~gh-temperature expansion of eq (4)

386

M Creutz, K J M Mortartv/ U(3)four-dtmenwonallatttce gauge theory

We would hke to thank the Science and E n g i n e e r m g Research Council of G r e a t Britain for the award of research grants (grants NG-0983.9 and NG-1068.2) to buy time on the C R A Y - l S c o m p u t e r at D a r e s b u r y L a b o r a t o r y where part of this calculation was carried out and the University of L o n d o n C o m p u t e r Centre for the g r a n t i n g of discretionary time on their C R A Y - I S where the rest of this calculation was performed. The calculations of the present paper took the equivalent of 88 C D C 7600 hours to perform. O n e of the authors ( K J M M ) wishes to thank the DESY directorate for the award of a Visiting Fellowshnp to visit DESY where this work was completed. Thns research was also carried out m part u n d e r the auspices of the US D e p a r t m e n t of Energy u n d e r contract no DE-AC02-76CH00016.

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Ftg 9 The smng tensnon~(I, J) for the U(I) component of pure U(3) gauge theory on a 64 lattice as a funcuon of the reverse temperaturc#. The full upward mangles represent (I, J) = (l, I), the full c]rclcs (2,2), the crosses (3,2) and the open orcles (3,3) Also shown m the diagram is the leading-order bagh-temperature expans]on of eq (5)

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References [ I ] M C r e u t z a n d K J M Monarty, Ph)s Rev D25(1982) 610 [2] M Creutz and K J M Monarty, Nucl Phys B210[FS6] (1982)59 [3] N Metropohs, A W Rosenbluth, M N Rosenbluth. A H Teller and E Teller. J Chem Phys 21 (1953) 1087 [4] M Creutz, 1981 Ence Lectures [5] K G Wdson, Phys Rev DI0 (1974) 2445 [6] M Creutz, Ph~s Rev Lett 43 (1979) 553 [7] M Creutz, Ph3,s Rev Lett 45 (1980) 313 [8} F Green and S Samuel, Phy,, Lett 103B (1981) 48, S Samuel, Calculating the large N phase transition m lattice gauge theories, Columbia Umver,,lty prepnnt. CU-TP-216, (October, 1981) and references thereto [9] M Creutz, L Jacobs and C Rebbl, Phy~ Rev Left 42 (1979) 1390 [10] B Lautrup and M Nauenberg, Phys Lett 95B (1980) 63. T A Degrand and D Tou,,smnt, Phys Re,, D24 (1981) 466, (} Bhanot, Phys Rev D24 (1981) 461, K J M Monarty, Phys Rev D25 (1982) 2185 [1 I] K J M Mormrty, Monte Carlo study of the stnng tension of compact U(I) gauge theory m four dimensions, Royal Holloway College prepnnt (Apnl, 1982)